Given a symmetric monoidal functor $\mathcal{C} \rightarrow \mathcal{C}'$, the property that $\mathcal{C}' \otimes_{\mathcal{C}} \mathcal{C}' \rightarrow \mathcal{C}'$ be an isomorphism is equivalent to $\mathcal{C} \rightarrow \mathcal{C}'$ being an epimorphism in the category of symmetric monoidal dg categories. This is a purely formal fact: in any cocartesian symmetric monoidal category $\mathcal{E}$ a map out of the initial object $0 \rightarrow Z$ is an epimorphism if and only if the map $Z \coprod Z \rightarrow Z$ is an isomorphism. Our case follows from this by taking $\mathcal{E}$ to be the category of symmetric monoidal categories over $\mathcal{C}$.
The three notions of localization mentioned in the question are epimorphisms, since a map out of the localization is by definition a map out of $\mathcal{C}$ satisfying a property (namely, that a certain arrow becomes invertible, that a certain object becomes invertible, etc). Hence all three notions satisfy the tensor idempotence condition.
Inverting an arrow $f: X \rightarrow Y$ amounts to passing to the quotient by the ideal generated by the cofiber of $f$. A symmetric monoidal functor $F: \mathcal{C} \rightarrow \mathcal{D}$ maps $\operatorname{cofib}(f)$ to zero if and only if it inverts $1_{\mathcal{C}} \oplus \operatorname{cofib}(f)$. Therefore your first notion of localization is a particular case of the second one.
Your third notion of localization is in fact equivalent to the second one. Given a map $f: X \rightarrow Y$ between invertible objects, any such map $f': X^{-1} \rightarrow Y^{-1}$ is necessarily dual to an inverse to $f$. Hence localizing in your third way along an arbitrary map $f:X \rightarrow Y$ is equivalent to first inverting $X, Y$ and then inverting $f$, which we already observed can be reduced to the second notion.
There are epimorphisms that do not arise as quotients by ideals: Consider for instance the category $\operatorname{Sh}(M)$ of sheaves of (complexes of) vector spaces on a manifold $M$. Let $x$ be a point in $M$ and $U$ its complement. The star pullback functor $\operatorname{Sh}(M) \rightarrow \operatorname{Sh}(\lbrace x \rbrace)$ exhibits $\operatorname{Sh}(\lbrace x \rbrace)$ as the quotient of $\operatorname{Sh}(M)$ by the ideal of sheaves with vanishing stalk at $x$. Similarly, $\operatorname{Sh}(U)$ is the quotient of $\operatorname{Sh}(M)$ by the ideal of sheaves supported at $x$. The ideal generated by the union of these two ideals is the whole $\operatorname{Sh}(M)$, so we see that $\operatorname{Sh}(\lbrace x \rbrace) \otimes_{\operatorname{Sh}(M)} \operatorname{Sh}(U) = 0$. It follows that the functor $\operatorname{Sh}(M) \rightarrow \operatorname{Sh}(\lbrace x \rbrace) \times \operatorname{Sh}(U)$ satisfies the tensor-idempotence condition, but it doesn't arise as the quotient by an ideal since its right adjoint $\operatorname{Sh}(\lbrace x \rbrace) \times \operatorname{Sh}(U) \rightarrow \operatorname{Sh}(M)$ (given by star-pushforward in each coordinate) is not fully faithful.
Under tameness conditions all notions of localization agree: I don't know if every epimorphism arises by inverting an object in general, but under certain tameness conditions one can show that this is the case:
Claim: Let $\mathcal{C}$ be a symmetric monoidal dg category compactly generated by its dualizable objects and $\mathcal{C}'
$ be a compactly generated symmetric monoidal dg category equipped with a symmetric monoidal functor $\mathcal{C} \rightarrow \mathcal{C}'$ that preserves compact objects, and such that the map $\mathcal{C}' \otimes_{\mathcal{C}} \mathcal{C}' \rightarrow \mathcal{C}'$ is an isomorphism. Then the functor $\mathcal{C} \rightarrow \mathcal{C}'$ arises by passing to the quotient by an ideal of $\mathcal{C}$.
Sketch of proof: Let $\mathcal{K}$ be the full subcategory of $\mathcal{C}'$ generated under colimits by the image of the functor $\mathcal{C}\rightarrow \mathcal{C}'$. Our conditions guarantee that the right adjoint to the inclusion $\mathcal{C} \rightarrow \mathcal{K}$ is colimit preserving and monadic. Note moreover that $\mathcal{K}$ is a $\mathcal{C}$-module and the functor $\mathcal{C} \rightarrow \mathcal{K}$ is a map of $\mathcal{C}$-modules. Its right adjoint in principle only commutes with the $\mathcal{C}$-action up to natural transformations, but the fact that $\mathcal{C}$ is compactly generated by its dualizable objects guarantees that the natural transformations are isomorphisms, and so the functor $\mathcal{K}\rightarrow \mathcal{C}$ is also a morphism of $\mathcal{C}$-modules. It follows that $\mathcal{K}$ is the category of algebras for a $\mathcal{C}$-linear monad on $\mathcal{C}$, and so we have an identification $\mathcal{K} = A\operatorname{-mod}$ for some algebra $A$ in $\mathcal{C}$. The fact that $\mathcal{C}' \otimes_{\mathcal{C}} \mathcal{C}' = \mathcal{C}'$ implies that the multiplication map $A \otimes A \rightarrow A$ is an isomorphism. This means that $\mathcal{K}$ is in fact the category of algebras for an idempotent $\mathcal{C}$-linear monad, and so it arises as the quotient of $\mathcal{C}$ by an ideal. This whole thing reduces you to understanding the case when $\mathcal{C} \rightarrow \mathcal{C}'$ is fully faithful. Since the canonical map $\mathcal{C}\otimes_{\mathcal{C}}\mathcal{C}' \rightarrow \mathcal{C}'\otimes_{\mathcal{C}} \mathcal{C}'$ is an isomorphism, we have that $\mathcal{C}'/\mathcal{C} \otimes_{\mathcal{C}} \mathcal{C}'$ vanishes. This contains $\mathcal{C}'/\mathcal{C} \otimes_{\mathcal{C}} \mathcal{C} = \mathcal{C}'/\mathcal{C}$ as a full subcategory, so we see that $\mathcal{C}' = \mathcal{C}$ is the trivial localization.
The algebro-geometric case: The above includes for instance the case of $\mathcal{C} = \operatorname{QCoh}(X)$ for $X$ a separated scheme. Moreover, from the proof we see that the resulting localizations are categories of modules for quasicoherent sheaves of algebras $A$ over $X$ such that the multiplication map $A \otimes A \rightarrow A$ is an isomorphism. In the Noetherian case you can use the result stated in the question to deduce that $A$ is locally a localization of the structure sheaf, so you see that all localizations in this sense are classified by collections of points of $X$ closed under specialization.
If you drop the condition that the functor $\mathcal{C} \rightarrow \mathcal{C}'$ preserves compact objects there are more examples, even in the geometric case. Indeed, any ideal of $\operatorname{QCoh}(X)$ provides an example, and these are classified (in the Noetherian case) by arbitrary collections of (non necessarily closed) points of $X$. This classification goes back to Hopkins, Neeman, and by now there is a whole industry about it - key words being tensor triangular geometry and classification of localizing subcategories.
Beyond algebraic geometry: If you are looking for interesting covers beyond Zariski covers in algebraic geometry one source could be topology. If you have a manifold $M$ and $U$ is an open set of $M$, the category $\operatorname{Sh}(U)$ is the category of comodules for an idempotent coalgebra in $\operatorname{Sh}(M)$ with underlying sheaf $k_U$, and so it can be obtained as the colimit
$$\operatorname{Sh}(M) \xrightarrow{\otimes k_U}\operatorname{Sh}(M) \xrightarrow{\otimes k_U} \operatorname{Sh}(M)\xrightarrow{\otimes k_U} \ldots .$$
This is the same diagram you would use to invert $k_U$, and so $\operatorname{Sh}(U)$ in fact results from inverting $k_U$. From this it follows that $\operatorname{Sh}(U)\otimes_{\operatorname{Sh}(M)} \operatorname{Sh}(V) = \operatorname{Sh}(U \cup V)$ for every pair of opens, and so you see that any open cover of $M$ provides a cover of symmetric monoidal categories in your sense.
You can build many more examples by variations of this theme. You could take $M$ to be the union of two manifolds $U, V$ along a closed submanifold (for instance $M$ could be the union of the $x$ and $y$ axes in $\mathbb{R}^2$) and you still have that $\operatorname{Sh}(M)$ is covered by $\operatorname{Sh}(U)$ and $\operatorname{Sh}(V)$. You could even require your sheaves to be constructible with respect to a stratification and $U, V$ to respect the stratification to get covers of categories of modules over quivers.
