Perhaps part of the issue is (to my perception) a confounding of three different, but related, issues. The second question asked about the dependence of Ext, Tor (and, presumably, other derived functors) on resolutions: the usual arguments not only show that the isomorphism classes of the Ext groups do not depend on the resolution, but that the (natural) maps among Ext groups of different objects are independent of the resolutions. One point is that this "naturality" is often confused with a more colloquial (and still useful) sense of "not making irrelevant choices", etc.
This is related to one of the earlier comments, that too many things were being identified by isomorphisms, and that various categories of finite-dimensional vector spaces already illustrate relevant issues. Namely, we can certainly consider the category with objects $k^0$, $k^1$, $k^2$, ... with a field $k$, that is, the elementary-linear-algebra finite-dimensional vector spaces over $k$. And we can consider a great variety of maps among these. As in the earlier comment, this is a "skeleton" of the category of all finite-dimensional $k$-vectorspaces. However, notice that we do not declare every isomorphism of $k^n$ with itself to be the identity.
The third point, mentioned in a comment, but which I suspect was not a primary part of the question(s) as asked, might indeed be about derived categories: how to make (e.g.) projective resolutions _modulo_homotopy_ the objects in a category, since we know that homotopic resolutions produce the same outcomes. My reaction is that the "natural" dependence of computing derived functors "up to isomorphism" should seem reasonable before worrying about derived categories.
The standard useful exercise in appreciating genuine naturality is to show that finite-dimensional vector spaces are definitely not naturally isomorphic to their duals... by showing that no list (over f.d. $k$-vectorspaces $V$) of isomorphisms $\phi_V:V\rightarrow V^*$ is compatible with all homs among vectorspaces. (And/but, equally standard is writing down the natural isomorphisms of these to their second duals.)