English grammar has no systematic category for "plural of plural" words. Fishes is a special, anomalous case, and it's optional, not mandatory, to use it in that context.
Historically, fishes did not originate as a plural of a plural. In formation, it's just an ordinary plural. The Old English source of fish is fisc, which had the plural fiscas. Eventually, there came to be a trend for animal words in English to have invariable plurals. This resulted in fish having two alternative plural forms, fishes and fish. And since fishes looks 'more plural', it seems like a lot of people feel like it must be the better form to use when talking about types of fish. But you can also just use the regular plural fish, even if you're talking about different species of fish.
And on the other hand, fishes, although the less common plural form, is not strictly limited to cases where you are talking about different kinds: the King James Bible (which of course is not always modern in usage) has examples like "We have here but five loaves, and two fishes". I don't advise imitating this wording; I only cite this as evidence that the form fishes is not fundamentally restricted to being used as a "plural of plural".
Deer has a somewhat different history. It comes from Old English deor, plural deor. Some alternative plurals developed in its history (the Oxford English Dictionary mentions "Middle English deore, deoran, Middle English deoren; Middle English deores, dueres, 1600s–1800s occasionally deers") but I don't think any was ever well established in standard modern English. Don't say "I saw three different kinds of deers."
Here are Google Ngram Viewer results for "types of":

