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Kan extensions make sense in any $2$-category. But so far I have only really seen them in the case of the $2$-category of categories, functors, natural transformations and the $2$-category of $k$-linear categories, $k$-linear functors, natural transformations. Where have Kan extensions been studied and used explicitly for other $2$-categories (a related question was asked here, but this doesn't really go into examples)? In particular I am interested in the $2$-category of monoidal categories, lax monoidal functors and lax monoidal natural transformations, also with "strong" instead of "lax". For example, the left Kan extension of a lax monoidal functor $(f,\eta,\mu) : I \to A$ (between monoidal categories $I,A$) along the unique lax monoidal functor $I \to \{1\}$ is a universal monoid object $M=(X,\eta,\mu)$ in $A$ equipped with a cocone $\{\alpha_i : f(i) \to X\}_{i \in \mathrm{Ob}(I)}$ which is lax monoidal in the sense that (a) $1 \xrightarrow{\eta} f(1) \xrightarrow{\alpha_1} X$ equals $1 \xrightarrow{\eta} X$ and (b) the diagram $$ \begin{array}{ccc} f(i \otimes j) & \xrightarrow{\alpha_{i \otimes j}} & X \\ {\scriptsize\mu}\uparrow ~~&&~~\uparrow{\scriptsize\mu} \\ f(i) \otimes f(j) & \xrightarrow{\alpha_i \otimes \alpha_j} & X \otimes X \end{array} $$ commutes. Is there any interesting example for this kind of "lax monoidal colimit"? If yes, what does it mean for a monoidal category to be "lax monoidal cocomplete"? Consider for example the case that the underlying category of a monoidal category is cocomplete and the tensor product is cocontinuous in each variable (this is what I would call a cocomplete monoidal category, or perhaps more precisely, a monoidal cocomplete category, because this is precisely a monoid in the symmetric monoidal $2$-category of cocomplete categories), which is quite typical. Do "lax monoidal colimits" exist then? What about "lax monoidal left Kan extensions" in general or "strong monoidal left Kan extensions" in general?

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I believe that this is a particular case of Lurie's "operadic left Kan extension". We may identify a monoidal $\infty$-category $\mathcal{C}$ with a coCartesian fibrations of $\infty$-operads $\mathcal{C}^{\otimes} \longrightarrow \mathcal{Ass}^{\otimes}$ where $\mathcal{Ass}^{\otimes}$ is the associative operad considered as an $\infty$-operad. If $\mathcal{A}^{\otimes} \longrightarrow \mathcal{Ass}^{\otimes}$ is another coCartesian fibration (i.e., another monoidal $\infty$-category $\mathcal{A}$), then the notion of a lax monoidal functor $\mathcal{A} \longrightarrow \mathcal{C}$ is given by the notion of a map of $\infty$-operads $\mathcal{A}^{\otimes} \longrightarrow \mathcal{C}^{\otimes}$ over $\mathcal{Ass}^{\otimes}$, and can be thought of as the data of an $\mathcal{A}$-algebra object in $\mathcal{C}$. For this reason the corresponding $\infty$-category of lax monoidal functors is sometimes denoted by $Alg_{\mathcal{A}}(\mathcal{C})$. Now if $\mathcal{A}^{\otimes} \longrightarrow \mathcal{B}^{\otimes} \longrightarrow \mathcal{Ass}^{\otimes}$ are maps of $\infty$-operads then the formation of operadic left Kan extension over $Ass^{\otimes}$ yields a left adjoint $$ LK: Alg_{\mathcal{A}}(\mathcal{C}) \longrightarrow Alg_{\mathcal{B}}(\mathcal{C}) $$ to the forgetful functor $Alg_{\mathcal{B}}(\mathcal{C}) \longrightarrow Alg_{\mathcal{A}}(\mathcal{C})$. We may hence think about it as associating to an $\mathcal{A}$-algebra $X$ the free $\mathcal{B}$-algebra generated from $X$. This construction is studied in section 3.1.3 of "Higher Algebra". A general existence result for free algebras is given by Corollary 3.1.3.5. It essentially says exactly what you propose in your question: if a monoidal category $\mathcal{C}$ has colimits and these are preserved by the tensor product in each variable separately then $\mathcal{C}$ admits the formation of free algebras (and in particular the formation of lax monoidal Kan extensions as you describe).

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This does not answer the specific questions you ask but is a paper which is focused precisely on computing monoidal Kan extensions:

Paul-André Melliès and Nicolas Tabareau. Free models of T-algebraic theories computed as Kan extensions. Preprint. (Presented at the International Category Theory Conference, Calais, 2008).

One of the main results is a sufficient condition for the usual end/coend formula (which computes the object part of Kan extensions in $\mathbf{Cat}$) to be also valid in $\mathbf{SymMonCat}$ (or just $\mathbf{MonCat}$ I guess). Their original motivation, if I remember correctly, is to understand the existence/non-existence of free commutative (co)monoids in a given category. Adapting Lawvere's approach to the monoidal case, a commutative monoid in a symmetric monoidal category $\mathcal C$ is a strict monoidal functor from the PROP $\mathbb F$ (finite ordinals and arbitrary functions between them) to $\mathcal C$, so the free commutative monoid on an object $A$ of $\mathcal C$ is the left Kan extension of $A$ (seen as a strict monoidal functor from the PROP $\mathbb B$ of finite ordinals and bijections) along the inclusion $\mathbb B\hookrightarrow\mathbb F$. However, the Kan extension must be computed in $\mathbf{SymMonCat}$ (with strict monoidal functors as $1$-cells), otherwise we do not even find a monoid. That's why the usual end formula may not work, so the authors try to explain when and why.

Of course, this not only fails to address your specific questions, but it may have nothing to do with what you need, because the above paper is mostly interested in strict monoidal functors. However, some of the definitions and results should work for strong functors too (I don't know about lax). The use of symmetric monoidal categories, on the other hand, is not essential (the reason for their interest in commutative (co)monoids is that, originally, the authors were trying to give an explicit formula for the free exponential modality $!A$ of linear logic, which is a free commutative comonoid). Anyway, I thought it would not hurt to give you this reference.

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Here are some miscellaneous remarks/thoughts on these notions.

Motivation for Monoidal Co/Limits and Variant Notions. (You've said part of this already, but let me start here for context)

If we want to replace the universal property $$\mathrm{Hom}_{\mathcal{D}}(-,\lim(D))\cong\mathrm{Nat}(\Delta_{(-)},D)$$ f the limit of a diagram $D\colon\mathcal{C}\to\mathcal{D}$ by a monoidal analogue such as $$\mathrm{Hom}_{\mathcal{D}?}(-,{\lim}^{\otimes}(D))\cong\mathrm{Nat}^{\otimes,\mathrm{lax}}(\Delta_{(-)},D)$$ ith now $D$ a lax monoidal functor and $\mathrm{Nat}^{\otimes,\mathrm{lax}}(F,G)$ the set of monoidal natural transformations from a lax monoidal functor $F$ to another lax monoidal functor $G$, then we have to look at lax monoidal structures on the constant functor $\Delta_{X}$ for $X\in\mathrm{Obj}(\mathcal{D})$. These turn out to be in bijection with monoid structures on $X$: $$\{\text{lax monoidal structures on $\Delta_{X}$}\}\cong\{\text{monoid structures on $X$}\}.$$ As such, instead of having a functor $$\Delta_{(-)}\colon\mathcal{D}\to\mathsf{Fun}(\mathcal{C},\mathcal{D})$$ given by the assignment $X\mapsto\Delta_{X}$, we now have a functor $$\Delta_{(-)}\colon\mathsf{Mon}(\mathcal{D})\to\mathsf{Fun}^{\otimes,\mathsf{lax}}(\mathcal{C},\mathcal{D})$$ given by the assignment $(X,\mu_{X},\eta_{X})\mapsto(\Delta,\Delta^{\otimes},\Delta^{\otimes,\mathbf{1}})$. (I hope the notation is clear here) This means our desired defining universal property should have the form $$\mathrm{Hom}_{\mathsf{Mon}(\mathcal{D})}(-,\lim(D))\cong\mathrm{Nat}^{\otimes,\mathrm{lax}}(\Delta_{(-)},D).$$

There are now two natural variants of the above notion:

  1. First, we may also consider commutative monoids. For $\mathcal{C}$ and $\mathcal{D}$ braided monoidal categories and $(X,\mu_{X},\eta_{X})$ a monoid in $\mathcal{D}$, the following conditions are equivalent:

    1. The lax monoidal functor $(\Delta_{X},\Delta^{\otimes},\Delta^{\otimes,\mathbf{1}})$ associated to $(X,\mu_{X},\eta_{X})$ is braided.
    2. $(X,\mu_{X},\eta_{X})$ is a commutative monoid in $\mathcal{D}$.

    As a consequence, the restriction of $\Delta_{(-)}\colon\mathsf{Mon}(\mathcal{D})\to\mathsf{Fun}^{\otimes,\mathsf{lax}}(\mathcal{C},\mathcal{D})$ to $\mathsf{CMon}(\mathcal{D})$ defines a functor $$\Delta_{(-)}\colon\mathsf{CMon}(\mathcal{D})\to\mathsf{Fun}^{\otimes,\mathsf{lax},\mathsf{br}}(\mathcal{C},\mathcal{D}),$$ and we may define the "commutative monoidal limit" of a braided lax monoidal functor $D\colon\mathcal{C}\to\mathcal{D}$ to be, if it exists, the commutative monoid $\lim^{\otimes,\mathsf{ab}}(D)$ in $\mathcal{D}$ such that we have a natural isomorphism $$\mathrm{Hom}_{\mathsf{CMon}(\mathcal{D})}(-,{\lim}^{\otimes,\mathsf{ab}}(D))\cong\mathrm{Nat}^{\otimes,\mathrm{lax},\mathrm{br}}(\Delta_{(-)},D),$$ where $\Delta_{(-)}$ is now a functor from $\mathsf{CMon}(\mathcal{D})$ to $\mathsf{LaxBrMonFun}(\mathcal{C},\mathcal{D})$, and the above is a natural isomorphism of functors from $\mathsf{CMon}(\mathcal{D})$ to $\mathsf{Sets}$.

  2. Instead of working with lax monoidal functors, we could just as well work with oplax ones. This leads to "universal comonoids, bimonoids, etc." via the bijections $$ \begin{align*} \{\text{oplax monoidal structures on $\Delta_{X}$}\} &\cong \{\text{comonoid structures on $X$}\},\\ \{\text{bilax monoidal structures on $\Delta_{X}$}\} &\cong \{\text{bimonoid structures on $X$}\},\\ \{\text{Hopf monoidal structures on $\Delta_{X}$}\} &\cong \{\text{Hopf monoid structures on $X$}\}. \end{align*} $$

In summary, we have really a number of different though closely related notions, such as "lax monoidal co/limits", "Hopf monoidal co/limits", "bicommutative Hopf monoidal co/limits", etc.

Formulas for Computing Monoidal Kan Extensions. In ordinary category theory, one may prove the formula $$\mathrm{Lan}_{K}(F)\cong\int^{A\in\mathcal{C}}\mathrm{Hom}_{\mathcal{D}}(K(A),-)\odot F(A)$$ by first defining ends, showing that $\int_{A\in\mathcal{C}}\mathrm{Hom}(F_{A},G_{A})$ computes $\mathrm{Nat}(F,G)$, and then doing a coend manipulation using the universal property of $\mathrm{Lan}_{K}(F)$ to arrive at the above.

It seems there's no such formula for monoidal Kan extensions: If we define "monoidal ends" via a universal property of the form $$\mathrm{Hom}_{\mathsf{Mon}(\mathcal{D})}\left(-,\int^{\otimes}_{A\in\mathcal{C}}D^{A}_{A}\right)\cong\mathrm{DiNat}^{\otimes,\mathrm{lax}}(\Delta_{(-)},D),$$ then it seems a formula such as $$\int^{\otimes}_{A\in\mathcal{C}}\mathrm{Hom}_{\mathcal{D}}(F_{A},G_{A})\cong\mathrm{Nat}^{\otimes,\mathrm{lax}}(F,G)$$ is false, as there is no reasonable way to put a monoid structure in the set $\mathrm{Nat}^{\otimes,\mathrm{lax}}(F,G)$.

So arguably it seems plausible/likely that a formula for monoidal Kan extensions of the form $$\mathrm{Lan}^{\otimes,\mathrm{lax}}_{K}(F)\cong\int^{A\in\mathcal{C}}_{\otimes}\mathrm{Hom}_{\mathcal{D}}(K(A),-)\odot F(A)$$ would not hold.

Weighted Monoidal Co/Limits. It also seems difficult to formulate monoidal variants of weighted co/limits. In a tentative universal property of the form $$\mathrm{Hom}_{\mathsf{Mon}(\mathcal{D})}(-,{\lim}^{\otimes,[W]}(D))\cong\mathrm{Nat}^{\otimes,\mathrm{lax}}(W,h^{-_{1}}_{D}),$$ we would have $W$ and $h^{-_{1}}_{D}$ lax monoidal functors from $\mathcal{D}$ to $\mathsf{Sets}$. However, to define a lax monoidal structure on $h^{-_{1}}_{D}$, we would need to supply morphisms of the form $$\underbrace{h^{-}_{D(A)}\times h^{-}_{D(B)}}_{\cong h^{-}_{D(A)\times D(B)}}\longrightarrow h^{-}_{D(A\otimes B)}.$$ By the Yoneda lemma, these are in bijection with morphisms $D(A)\times D(B)\longrightarrow D(A\otimes B)$ of $\mathcal{D}$, whereas the lax monoidal structure on $D$ gives us a morphism of the form $D(A)\otimes D(B)\longrightarrow D(A\otimes B)$.

Edit: This is not a problem, we just need to use the Day convolution monoidal structure $\circledast$ on $\mathsf{PSh}(\mathcal{C})$, which provides us with the necessary isomorphisms $h_{A}\circledast h_{B}\cong h_{A\otimes_{\mathcal{C}}B}$. So one can define weighted monoidal co/limits via universal properties of the form $$\mathrm{Hom}_{\mathsf{Mon}(\mathcal{D})}(-,{\lim}^{\otimes,[W]}(D))\cong\mathrm{Nat}^{\circledast,\mathrm{lax}}(W,h^{-_{1}}_{D}).$$

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I am not an expert, but I think the best reference for this topic is Algebraic Kan extensions along morphisms of internal algebra classifiers by Mark Weber.

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