I don't know whether this question makes complete sense, but I'm 90% certain it does.
In the definition of a coalgebra over a field, the fact that $(C \otimes C) \otimes C \cong C \otimes (C \otimes C)$ is a natural isomorphism enables the two to be equated. But do we need the naturalness, or can we settle for an unnatural isomorphism?
Similarly, the duality functor on $k$-FinVec reverses the directions of all the arrows (because it's contravariant). Write the duality functor as $(-)^*$. Does the fact that the isomorphism $(C \otimes C)^* \cong C^* \otimes C^*$ is natural play any important role in the fact that every finite-dimensional coalgebra is dual to a finite-dimensional algebra. If we drop naturalness, then $(C \otimes C)^* \cong C^* \otimes C^*$ is true simply because of dimension counting. So why can't we use the existence of an (unnatural) isomorphism alone? Does it have something to do with the morphisms of the form $\operatorname{id}\otimes \Delta$, and the way that $\otimes$ is combining two morphisms together?
In general, one can define what a coalgebra is on any tensor category. Such a thing is an abelian category $C$ endowed with a bifunctor $\otimes:C\times C\to C$, such that there is an object $\mathbf1$, and natural isomorphisms $a:(X\otimes Y)\otimes Z\to X(\otimes Y\otimes Z)$, $l:X\otimes\mathbb1\to X$, and $r:\mathbb1\otimes X\to X$ that satiasfy certain conditions. Notice that these maps are what you need to define a coalgebra, as you wrote.
The point is that an abelian category may have several different tensor structures, tuples $(\otimes,\mathbb1,a,l,r)$ that satisfy the needed conditions, and it may well be the case that what being coalgebra in $C$ with respect to a tensor structure be very different from what a coalgebra in $C$ with respect to another tensor structure.
It is very often the case that on an abelian category $C$ there are two tensor structures $(\otimes,\mathbb1,a,l,r)$ and $(\otimes',\mathbb1',a',l',r')$ which only differ in that $a$ and $a'$ are different, for example. That is why the map $a$, called the associator, is important.
It is not difficult to construct examples of this. For example, the category $C$ of vector spaces endowed with a grading over a group $G$ has many different tensor structures corresponding to $3$-cocycles $G\times G\times G\to K^\times$ with values in the multiplicativ group of the ground field. This is treated in textbooks that have «tensor categories» or «monoidal categories» in their title.