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It's a matter of historical origin and subsequent development.
In the oldest recorded English deer belonged to the neuter declension, which did not have a distinct plural ending in the nominative and accusative cases. (It is believed that this declension did have plurals in Proto-Germanic, but they disappeared before English or any immediate ancestor was written down.) At that time there was no ambiguity, since the determiners accompanying these nouns did change in the plural.
Later, when the Old English endings were mostly lost, the majority of these neuter nouns acquired 'regular' plural endings in -n, eventually superseded by endings in -s: wīf, for instance, became wives in the plural. A few, however did not, and deer is one of these.
It is often remarked that all these nouns with invariant plurals denote animals, deer, sheep, fish, swine, which are either herded or hunted; and it has been suggested that both the 'mass noun' sense with herd animals and the custom of referring to all hunted animals in the singular (we hunt bear, lion, and elephant as well as deer) helped inhibit plural regularization.
ADDED: See the second edition (1954) of Jespersen, A Modern English Grammar on Historical Principles, Part II SYNTAX (First Volume), Ch.III The Unchanged Plural (pp. 49–69), especially 3.1–3.2 and 3.71.
A good answer of StoneyB. I can only add that the lack of distinction between plural and singular forms of some old nouns (which logically must have this distinction) exists in many languages and can be traced back to the ancient state of the language, where the same word was used to describe both the class of elements and one particular element. For example, such a peculiarity still can be found in Korean or Chinese - you usually don't bother about plural ending, unless you want to emphasize the plurality.
Deers has been used that way.
From A Popular History of Mammalia, 1850:
The subgenus Muntjacus is distinguished from the other deers by the horns being supported on elongated pedicels ...
From Comparative Genomics, 2000:
In contrast to other deers, the muntjaks display chromosome numbers ranging from the lowest chromosome number known in mammals of 2n=6 (female) and 2n=7 (male Indian muntjak; Muntiacus muntjak vaginalis. Figure 8) to 2n = 46 (Chinese muntjak; Muntiacus reeversi).
I would say that either deers or deer would be acceptable for this type of usage. However, if you want to be indisputably correct, you could use species of deer. I also suspect this is the most common term for this meaning, as neither deer nor deers sounds quite correct to me here.
I'm afraid that "deers" is not correct, which is to say that the form has never been in common use in the sense you suggest, as far as the OED is aware.
As you say, it's common today to use "fish" as the plural of "fish" (e.g. "two fish"), and certain people insist that you must only use "fishes" when talking about different fish species. But the original form "fishes" was simply the plural of fish (e.g. "two fishes"), and didn't necessarily mean two species. A form ending in "s" occurs in the Old English Vespasian Psalter around 825 ("Fuglas heofenes & fiscas saes"), while the collective singular form "fish" used for the plural does not occur until the fourteenth century (OED s.v. 1b). The plural "fishes" is still frequently found alongside "fish", and the rule about using "fishes" to refer only to species of fish is probably a recent development to make sense of this redundancy. At any rate, the OED does not recognize a unique sense of fish meaning "species of fish", whose plural "fishes" distinguishes it from the more familiar noun with the plural "fish". It certainly is not usual in English that a word denoting an animal should have two alternate plurals, the one referring to multiple instances of one species, the other referring to multiple species.
You cannot use "deers" to mean "species of deer" because "deers" has never been a form in common use, whereas "fishes" has been and continues to be. For that reason "deers" grates on the ear of a native speaker in a way that "fishes" doesn't.
The form "deers" is occasionally (though very rarely) found: Hogg's Tales and Sketches (1817) speaks of a "place of rendezvous, to which the deers were to be driven." But none of the very few examples of "deers" given by the OED uses the form to indicate that species of deer are meant.
(Source: OED, s.v. "fish" n.1 and "deer" n.)