Why is the plural version of deer identical to the singular version?
If mouse became mice, then why did the singular deer not change to something else in the plural?
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endings were mostly lost, the majority of these neuter nouns acquired 'regular' plural endings in -n, eventually superseded by endings in -s: wīf, for instance, became wives in the plural.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.
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.