Let me give you some general thoughts, although I am not answering the question directly. As it was pointed out, quotient and localization are not similar constructions. Although, usually quotient categories are used to deal with localization problems - for example, in model category theory (where you construct a homotopy category (a special quotient category) which is equivalent to the localized category), or even in direct constructing a localization of a small category (as the localization of $ \mathbb{N} $).
To see the difference, you may observe the following: For simplicity, consider only functors bijective onto the objects and small categories. You can see that all such fully faithful functors could be seen as a quotient functor (projection of a category in a quotient category).
But they are not always a localization!
On the other hand, if you think about categories in general, the unit component/projection $X\to Localization (X) $ is not fully faithful in general, while the quotient functors are always fully faithful.
Quotients, in general (in mathematics), are special types of colimits: this is the case, for instance, for groups, topology and for categories. That said, you can conjecture that this is the reason why such "quotients" are well used to construct other types of colimits or images of left adjoint functors.
Concerning about your question, although quotient categories are somehow similar to quotient of groups (as I said in the last paragraph), quotient of categories is not a horizontal categorification of the quotient of groups. Rather, I would say that quotient categories may be seen as a horizontal categorification of quotient monoids.
And if you take a look at quotient monoids, it cannot be seen easily as quotient in the case of groups. I mean, precisely, if I am not mistaken right now, quotient of monoids can't be controlled by the inverse image of the unit/identity (as you do with groups). Take, for instance, the morphism $ \mathbb{N}\ast \mathbb{N}\to \mathbb{N}\times \mathbb{N} $, in which $ \ast $ denotes the coproduct (free product).
This morphism, by definition, is induced by the two different canonical inclusions $ \mathbb{N}\to \mathbb{N}\times \mathbb{N} $.
This morphism is a projection of a quotient (quotient of categories): however the inverse image of the identity is trivial.
As I said, the best analogy for localization in dimension $0$ seems to be localization of rings (or localization of monoids). So, my thought is that you can consider the localization as a horizontal categorification of localization of monoids.
At last, formally we have the following: consider the category $ sGpd $ of groupoids posets - or the category of groupoids such that each connected component is simply connected. A category with a suitable relation is just a $ sGpd $ - enriched category. We have a canonical inclusion $ U: CAT\to sGpd $ induced by the inclusion $ Set\to sGpd $ which sends each set to the discrete groupoid. If you have a Gpd-category $X$, its quotient is just its left 2-reflection along $U$.
On the other hand, to consider localization in general, it seems harder. But avoiding technical problems, you may consider the 2-category $W$ of categories endowed with (suitable) subcategories of weak equivalences and functors preserving such subcategories (and natural transformations). Again, there is a canonical inclusion $ T: CAT\to W $ which sends each category to the category endowed with the subcategory of isomorphisms. And the localization of a category $ X $ w.r.t. a suitable subcategory would be (if it exists) the left 2-reflection along $T$ of $X$ (considering $X$ as an object of $W$).
