The ABC of categories: ABstract vs Concrete From Wikipedia:

A concrete category is a category that is equipped
with a faithful functor to the category of sets.
This functor makes it
possible to think of the objects of the category as sets with
additional structure, and of its morphisms as structure-preserving
functions.

This definition gives rise to the well-established dichotomy between concrete and abstract categories.
The examples of abstract  (= non-concrete) categories I've heard of come in two flavours:

*

*categories with structured sets as objects, and some other sort of structure-preserving maps than standard homomorphisms, e.g. interpretations. (I understand that - basically - interpretations are structure-preserving maps, just involving re-definitions of individuals (as n-tupels) and relations.)


*categories with structured set as objects, and equivalence classes of structure-preserving maps as morphisms, e.g. the homotopy category of topological spaces or much simpler: the category $C'$ which collapses all arrows $X \rightarrow Y$ of a (concrete) category $C$ into one (what's its name?)
[Side remark: The trivial equivalence relation on morphisms: $f \sim g :\equiv f = g$ leaves a given category $C$ unchanged.]
Given such a vast variety of possible definitions of structure-preserving maps and equivalence relations between them, I wonder why only classical homomorphisms and the trivial equivalence relation give rise to so-called concrete categories?
The other way around:

What is an example of an abstract category (in the
standard sense) with structured sets as objects, such that we cannot think of its morphisms as some equivalence class
of some sort of structure-preserving maps?

 A: The smallest example would be the category whose only object is a 1-element set $A$ (structured just as a set, or, if you don't like that, as a topological space), with two morphisms, the identity $e$ and another morphism $f$, with the composite $ff$ equal to $e$.  We can't think of the morphisms as equivalence classes of structure-preserving maps $A\to A$, because there's only one map from $A$ to itself.  
I suspect this is not what you were really looking for, but then you should clarify what you had in mind --- perhaps you want to allow the possibility of ignoring the given structured sets and replacing them with others?
A: Example: The category whose objects are sets and whose morphisms are relations.
Example: The cobordism category of $n$-manifolds.
Example: The opposite category of your favorite concrete category.
There is certainly no obvious way to think of these as categories of sets with morphisms a slight generalization of standard homomorphisms. There might be some way to construct a  set-with-morphism interpretation but it seems unlikely to actually be a useful thing to do.
Certainly, when we do category theory on them, it is useful to think of them as structured sets with homomorphisms. Indeed, that is why they are called "objects" and "morphisms" instead of, say, "vertices" and "edges" - because it is useful to think of nearly any category as consisting of a (perhaps very strange) sort of structured set and a (perhaps very strange) sort of homomorphism. But this is just a process of pretending, that allows us to take intuitions from the common and well-studied sets and morphisms situation and apply them to stranger sorts of categories.
A: This article by Ivan di Liberti and Fosco Loregian delves deeply into the relationship between concreteness and homotopy:
Homotopical algebra is not concrete. On the concreteness of certain model categories
It is introduced by a quotation from Freyd:

A: A theorem of Kučera asserts that every category arises as a quotient of a concrete category by suitable congruence: see Section 6B of Theory of Mathematical Structures by Adámek.
