Yes, this usage can be heard. However, in North American English, at least, it would be more common and comprehensible to say "types of fish" or "kinds of fish".
Note that some people use "fishes" as the plural of "fish"anyway, and they raise eyebrows, but it's not impossible. I wonder if this is what makes this usage sound plausible, whereas I would find "sheeps" for "kinds of sheep" very strange.
There are some nouns this seems to work better for this operation than others. For example, "five meats", based on the mass noun "meat", clearly means "five kinds of meat". But "five beers" could mean five bottles or five kinds — even though "beer" is also originally a mass noun. In most cases, you can just add "different" and talk about "five different beers" to be clearer.
"Five different fish", on the other hand, is more ambiguous. I picture five individual fish, whether the same kind or not. (Like saying that five different people called you.) "Five different fishes" just sounds questionable. But "five (different) kinds of fish" is clear.