"Let Top be the category of topological spaces." If I see a definition like this, in which homeomorphic (isomorphic in the category) spaces are not identified together, then for each given topological space, how many times does it show up as an object in Top? Once? Countably many times? Uncountably many times? Is there a semantic crisis if we don't identify all homeomorphic topological spaces to the same object?
Well, such categories are actually proper classes. For instance, spaces that are homeomorphic to $\mathbb{R}$ will be given on ANY set of the same cardinality. So how many there are, depends on how many such sets there are...but there are quite a lot of sets. In fact, more than any individual set worth. (Though on the details there, I'm not sure about what happens for fixed cardinality) We CAN identify them, and we get a skeleton. However, we usually don't do this, because there's no natural way to pick one object in each class, and one of the important catch phrases for category theory is "never make a choice." (Naturally, this is a vast oversimplification, but it's a good rule of thumb) 


If what you mean by "the category of topological spaces" is the category whose objects are pairs $(S,T)$ where $S$ is a set and $T$ is a topology on $S$ and by topological space you mean a pair $(S,T)$ where $S$ is a set and $T$ is a topology on $S$, then clearly each topological space appears exactly once. 


It would be best to talk about the category of sets first, I think. Any isomorphism class of sets shows up so many times that a given isomorphism class doesn't itself form a set  for example, $\{ 1, 2, 3 \}$ or $\{ 4, 5, 6 \}$ or $\{ \text{Cat}, \text{Dog}, \text{Monkey} \}$ are all elements of the isomorphism class of sets of cardinality $3$, and more generally so is any set whose elements are three different sets, so there are at least as many sets of cardinality $3$ as there are sets. The same occurs in any category where you insist on representing the objects as sets, but one of the nice things about category theory is that it more or less doesn't matter what "machine language" you use to describe a category. 


This is not essentially different from what everyone else has said, but for some reason I feel like saying it anyway. The number of times that a given topological space appears in the category of topological spaces is exactly once. That's the definition of the class of objects of Top. As for isomorphism classes: there is exactly one object in Top which is isomorphic to the empty set $\varnothing$ with its unique topology: namely $\varnothing$. For any nonempty topological space $X$, the subclass of Top consisting of spaces $X'$ which are homeomorphic to $X$ forms a proper class: i.e., they are more numerous than any set. This is true because the class of all sets is a proper class, and if $(X,\tau)$ is any nonempty topological space and $S$ is any set, then $X \times \{S\}$ is a set which is different from $X$ but in bijection with it: $x \mapsto (x,S)$, and the image of $\tau$ under this bijection gives a topological space which is homeomorphic to $(X,\tau)$ but with a different underlying set. So, except for one, the homeomorphism classes in Top are "unsetically" huge. Nevertheless, the modern perspective is that this is fine (or, perhaps, harmlessly irrelevant) whereas it is a bad idea to work with the homeomorphism class of the space instead of the space itself. One way to think about this is that a given topological space $X$ is a relatively simple object, but the class of all topological spaces homeomorphic to $X$ is a ridiculously complicated object. (This has not always been the received wisdom: notably Bertrand Russell's definition of the number $2$ was the class of all sets which can be put in bijection with, say, $\{\emptyset, \{ \emptyset\} \}$.) From a less philosophical perspective, one wants to speak about sets of maps between two topological spaces $X$ and $Y$, and if $X$ and $Y$ are only welldefined up to a homeomorphism, these sets are themselves not so well defined. As soon as one studies commutative diagrams of morphisms of objects in a category (or more generally, functorial constructions), one recognizes that the concept of "topological space up to a homeomorphism" is a painfully awkward one. It is also simply against the spirit of category theory to pass to isomorphism classes: many would say that this overemphasizes the somewhat quaintly philosophical notion of equality of objects. Instead of saying "the topological space $X$ is equal to the topological space $Y$", many mathematicians now think it is both simpler and more useful to say "$\Phi: X \rightarrow Y$ is a homeomorphism". For more on this point, I highly recommend Barry Mazur's article When is one thing equal to some other thing? 


(Expanding on Yuan's point about the irrelevance of "machine language" and JT's about "the wrong question to ask":) In a sense, one oughtn't even be able to ask whether two topological spaces (i.e., objects of Top) are equal, only whether they are isomorphic; if one can't ask about equality, then one certainly can't speak of cardinality (with respect to such equality), and so there is no notion of the cardinality of the collection of objects isomorphic to a given one. However, if you construe every topological space as implicitly carrying extra nontopological information via which such a nontopological notion of equality is defined (e.g., if you take the points of topological spaces to furthermore be elements of the cumulative hierarchy of wellfounded sets of sets of sets..., allowing one to ask whether points in distinct spaces are equal by appeal to this extra structure, and accordingly whether spaces themselves are equal by virtue of an isomorphism sending points to equal points), then, of course, the question can be answered (in the given example, as noted above, the answer will be that the isomorphism classes form proper classes). But this is not really a question about the category of topological spaces, as such; this is a question about the particular manner in which one may choose to realize the intuitive theory of topological spaces within the ontology of another (meta)theory/implement the structure of the category of topological spaces within a context imposing further structure as well. If one avoids selecting such "implementation details", then the question is meaningless, in precisely the same way as questions such as "Is the integer 9 an element of the rational 3/5?" are meaningless in the abstract. 

