It is not true in general that the abelian subcategory (by which I mean sub-abelian category) generated by an object $X$ is all subquotients of finite sums of $X$. It is contained in these subquotients, but it might not be all of them. This is because, for instance, not every subobject of $X$ is the kernel of an endomorphism of $X$ (or more generally, a map from $X$ to a sum of copies of $X$).
As an example, consider the abelian subcategory of $\mathbb{Z}$-modules generated by $\mathbb{Q}$. Because any $\mathbb{Z}$-homomorphism of $\mathbb{Q}$-vector spaces is automatically $\mathbb{Q}$-linear, you get only the finite dimensional $\mathbb{Q}$-vector spaces, and don't, for instance, get the subgroup $\mathbb{Z}$ of $\mathbb{Q}$.
I'm not entirely sure what you're looking for in an answer, but maybe I'll flesh out my comment.
It looks like what you're describing is equivalent to the homotopy category associated to the model structure on Cat where the weak equivalences are equivalences of categories. (I can say "the" because there is only one such, as pointed out in the comments. The cofibrations are functors injective on objects, and the fibrations are "isofibrations".)
I would say that in this context your category has been much studied. In particular, it is interesting to ask questions about homotopy limits and colimits in this category because many useful constructions arise in this way. (Homotopy (co)limits with this model structure are the same as "2-(co)limits" which is the name appearing in most of the literature, especially older literature.)
An example application of this language is the following theorem: The subcategory of presentable (resp. accessible) categories is closed under homotopy limits.
Using this one can prove that most of your favorite things are presentable (resp. accessible). For example, the category of modules over a monad arises via a homotopy limit construction, and this takes care of most things of interest.
Here's a neat application of this (which is the ordinary category version of a result that can be found, for example, in Lurie's HTT, 5.5.4.16.).
Say you want to localize a category $\mathcal{C}$ with respect to some collection of morphisms, $S$. Usually $S$ will not be given as a set, but if $\mathcal{C}$ is presentable you're usually okay if $S$ is generated by a set. Well, it turns out that if $F: \mathcal{C} \rightarrow \mathcal{D}$ is a colimit preserving functor between presentable categories, and $S$ is a (strongly saturated) collection of morphisms in $\mathcal{D}$ that is generated by a set, then $f^{-1}S$ is a (strongly saturated) collection of morphisms generated by a set. The argument goes by way of showing that the subcategory of the category of morphisms generated by $f^{-1}S$ is presentable, using a homotopy pullback square.
Adapting this to the model category or $\infty$-category setting, one sees immediately that localizing with respect to homology theories is totally okay and follows formally from this type of argument. (Basically, after fiddling around with cells to prove the category of spectra is presentable, you don't have to fiddle any more to get localizations. This is in contrast to the usual argument found in Bousfield's paper. You've moved the cardinality bookkeeping into a general argument about homotopy limits of presentable categories.)
Anyway, apologies for the very idiosyncratic application of this language; these things have been on my mind recently. I'm sure there are much more elementary reasons why one would care about using the model category structure on Cat.
Best Answer
Do you want the notion of "subcategory" to be invariant under categorical equivalence? If so, then "pseudomonic" functors are the right thing: faithful, and full on isomorphisms.
But I don't think one would want this any more than the notion of "inclusion" of topological spaces to be invariant under homotopy equivalence (which would make it meaningless).