It might be worth to first consider the particular case of the symmetric monoidal category $({\rm Set},\times)$ of sets and Cartesian products. Let us mildly extend the setting to include possibly multi-colored operads. Then your construction becomes a particular case of the construction which associates to a category $\mathcal{C}$ the colored operad $\mathcal{C}^{\coprod}$ whose colors are the objects of $\mathcal{C}$ and such that for every finite set $I$, every $I$-tuple of objects $\{x_i\}_{i \in I}$ of $\mathcal{C}$ and every object $y \in \mathcal{C}$ the set of multi-operations $\{x_i\}_{i \in I} \to y$ is the set $\prod_i{\rm Hom}(x_i,y)$ (in other words, a multi-operation from $\{x_i\}_{i \in I}$ to $y$ is given by a collection of maps $f_i:x_i \to y$ for $i \in I$). The operad $\mathcal{C}^{\coprod}$ has several interesting features:

(1) If the category $\mathcal{C}$ has finite coproducts then $\mathcal{C}^{\coprod}$ is the underlying operad of the symmetric monoidal category $(\mathcal{C},\coprod)$.

(2) The operad $\mathcal{C}^{\coprod}$ is equivalent to the Boardman-Vogt tensor product of $\mathcal{C}$ (considered as a colored operad with no non-1-ary operations) and ${\rm Com}$. As a result, the notion of a $\mathcal{C}^{\coprod}$-algebra in a symmetric monoidal category $(\mathcal{D},\otimes)$ is equivalent to that of a $\mathcal{C}$-indexed diagram of commutative algebra objects in $\mathcal{D}$. More generally, if $\mathcal{O}$ is any operad then the category of operad maps $\mathcal{C}^{\coprod} \to \mathcal{O}$ is equivalent to the category of functors from $\mathcal{C}$ to the category of operad maps ${\rm Com} \to \mathcal{O}$.

(3) If $\mathcal{O}$ is a unital colored operad (that is, each color has a unique $0$-ary operation) then operad maps $\mathcal{O} \to \mathcal{C}^{\coprod}$ are the same as maps from the underlying category of $\mathcal{O}$ (obtained by taking only the $1$-ary operations) to $\mathcal{C}$.

In particular, the operad $\mathcal{C}^{\coprod}$ enjoys both a mapping property out of it and a mapping property into it (at least for unital operads), both of which characterize it up to a unique isomorphism. There is also a similar story in the setting of $\infty$-operads, where $\mathcal{C}$ is allowed to be any $\infty$-category, and the $\infty$-categorical analogue of all three properties above holds.

Returning to the single colored case where $\mathcal{C}$ is taken to be a monoid rather than a category we can still formulate (2) above as giving a characterization of the notion of an algebra object over $\mathcal{C}^{\coprod}$ in a general symmetric monoidal category. Your construction is a generalization of this case in another direction, where you replace your monoid (which can be considered as a bialgebra in the symmetric monoidal category $({\rm Set},\times)$) by a bialgebra $A$ in a general symmetric monoidal category $\mathcal{V}$. In this case, you can still characterize $\mathtt{P}_A$ as above by specifying what are $\mathtt{P}_A$-algebras in a given symmetric monoidal $\mathcal{V}$-enriched category $\mathcal{D}$: these will correspond to commutative $A$-algebra objects in $\mathcal{D}$, that is, objects equipped with a commutative algebra structure and an $A$-module structure which are compatible with each other. This compatibility can be formulated using the structure of $A$ as a bialgebra. For example, you can describe this structure as being a commutative algebra object in the category of $A$-modules, where the latter has a symmetric monoidal structure induced by the coproduct of $A$.

One might summarize this discussion as follows: the "reason" why there is a natural operad $\mathtt{P}_A$ is that for a bialgebra $A$ there is a natural notion of a commutative $A$-algebra, and so we can expect to have an associated operad which encodes this theory.

It might be worth to first consider the particular case of the symmetric monoidal category $({\rm Set},\times)$ of sets and Cartesian products. Let us mildly extend the setting to include possibly multi-colored operads. Then your construction becomes a particular case of the construction which associates to a category $\mathcal{C}$ the colored operad $\mathcal{C}^{\coprod}$ whose colors are the objects of $\mathcal{C}$ and such that for every finite set $I$, every $I$-tuple of objects $\{x_i\}_{i \in I}$ of $\mathcal{C}$ and every object $y \in \mathcal{C}$ the set of multi-operations $\{x_i\}_{i \in I} \to y$ is the set $\prod_i{\rm Hom}(x_i,y)$ (in other words, a multi-operation from $\{x_i\}_{i \in I}$ to $y$ is given by a collection of maps $f_i:x_i \to y$ for $i \in I$). The operad $\mathcal{C}^{\coprod}$ has several interesting features:

(1) If the category $\mathcal{C}$ has finite coproducts then $\mathcal{C}^{\coprod}$ is the underlying operad of the symmetric monoidal category $(\mathcal{C},\coprod)$.

(2) The operad $\mathcal{C}^{\coprod}$ is equivalent to the Boardman-Vogt tensor product of $\mathcal{C}$ (considered as a colored operad with no non-1-ary operations) and ${\rm Com}$. As a result, the notion of a $\mathcal{C}^{\coprod}$-algebra in a symmetric monoidal category $(\mathcal{D},\otimes)$ is equivalent to that of a $\mathcal{C}$-indexed diagram of commutative algebra objects in $\mathcal{D}$. More generally, if $\mathcal{O}$ is any operad then the category of operad maps $\mathcal{C}^{\coprod} \to \mathcal{O}$ is equivalent to the category of functors from $\mathcal{C}$ to the category of operad maps ${\rm Com} \to \mathcal{O}$.

(3) If $\mathcal{O}$ is a unital colored operad (that is, each color has a unique $0$-ary operation) then operad maps $\mathcal{O} \to \mathcal{C}^{\coprod}$ are the same as maps from the underlying category of $\mathcal{O}$ (obtained by taking only the $1$-ary operations) to $\mathcal{C}$.

In particular, the operad $\mathcal{C}^{\coprod}$ enjoys both a mapping property out of it and a mapping property into it (at least for unital operads), both of which characterize it up to a unique isomorphism. There is also a similar story in the setting of $\infty$-operads, where $\mathcal{C}$ is allowed to be any $\infty$-category, and the $\infty$-categorical analogue of all three properties above holds.

Returning to the single colored case where $\mathcal{C}$ is taken to be a monoid rather than a category we can still formulate (2) above as giving a characterization of the notion of an algebra object over $\mathcal{C}^{\coprod}$ in a general symmetric monoidal category. Your construction is a generalization of this case in another direction, where you replace your monoid (which can be considered as a bialgebra in the symmetric monoidal category $({\rm Set},\times)$) by a bialgebra $A$ in a general symmetric monoidal category $\mathcal{V}$. In this case, you can still characterize $\mathtt{P}_A$ as above by specifying what are $\mathtt{P}_A$-algebras in a given symmetric monoidal $\mathcal{V}$-enriched category $\mathcal{D}$: these will correspond to what one might call commutative $A$-algebra objects in $\mathcal{D}$, that is, objects equipped with a commutative algebra structure and an $A$-module structure which are compatible with each other. This compatibility can be formulated using the structure of $A$ as a bialgebra. For example, you can describe this structure as being a commutative algebra object in the category of $A$-modules, where the latter has a symmetric monoidal structure induced by the coproduct of $A$.

One might summarize this discussion as follows: the "reason" why there is a natural operad $\mathtt{P}_A$ is that for a bialgebra $A$ there is a natural notion of a commutative $A$-algebra, and so we can expect to have an associated operad which encodes this theory.