Proving Non-Concreteness of a Category
There is an important terminological discrepancy here: some authors use the term "concrete" for what others refer to as "concretizable" (see e.g. nLab versus Freyd). After a certain amount of exposure, this discrepancy is fairly benign, but initially it can be confusing. Contra the original version of this answer, I'll follow the nLab language.
Despite the name, a concrete category isn't just a category - it's a category together with a particular faithful functor into Sets. A concretizable category, meanwhile, is a category such that there exists such a functor (but we don't pick out a specific one).
So the linked post is correct in claiming that the category in question is not a concrete category - for the silly reason that it's just a category, as opposed to a category + a specific faithful functor into Sets. On the other hand, it's easy to show that it is concretizable.
- It's worth observing that all non-concretizable categories are pretty complicated (and in particular, all small categories are concretizable).
That said, the post linked is still slightly imprecise when it comes to defining concrete categories:
[C]oncrete categories [...] are categories where the objects are sets, usually with some additional structure (group structure, a topology, etc.), and the morphisms are well-defined functions between those sets that preserve the structure.
(Emphasis mine.) The bolded "are" isn't really accurate; rather, a concrete category is intuitively a category together with an "interpretation" of the objects and morphisms as sets and functions. But the objects aren't required to literally be sets, they just need to correspond to sets in some explicit way (namely, via a specific choice of faithful functor).
Call this category $\mathbf{C}$. Then, there are lots of functors from $\mathbf{C}$ to $\mathbf{Set}$, and they are all faithful because $\mathbf{C}$ is a poset.
The naive homotopy category of pointed topological spaces is not conrete. Here is how we construct it: $\textbf{Top}_*$ is the category of pointed topological spaces and continuous basepoint preserving functions between them.
Let $\text{NH}(\textbf{Top}_*)$, the naive homotopy category of pointed topological spaces be the category with the same objects as $\textbf{Top}_*$ but the hom sets $\text{Hom}_{\text{NH}(\textbf{Top})}((A,a),(B,b))$ are pointed homotopy classes of maps $(A,a) \rightarrow (B,b)$. I.e two maps are in the same class if there is a pointed homotopy between them.
Basic homotopy theory tells us that the usual composition of maps gives a well defined composition of maps on homotopy classes. I.e if we let $[f]$ be the set of pointed homotopy classes then the operation $[f] \circ[g] = [f \circ g]$ is well defined.
The proof that $\text{NH}(\textbf{Top}_*)$ is not concrete is complicated but you can read it here