In the beginning of the Torah, the Torah describes various min - types or categories of animals. Each has a distinct definition - animals, birds, fish etc. The defining feature of all bird-like animals is that they fly and have wings. They come from the water but that's the definition. So if you find a winged creature that you don't know how to classify, you have to know if it flies and has wings. The question is really what animal has the potential to fly, and has wings - but doesn't - what is it's status. So an Ostrich is one such example. Then there is the duck-billed platypus that lays eggs is another example of an in-between species.
Rashi (Lev. 11:13) explicitly says that a single min includes various different species.
Acc. to Menachos 29a, it seems even Moshe Rabbeinu was not clear on which animal and species was what without prophetic help.
Often different types of animals share the same name because they have similar characteristics. (for example התנשמת) used in Lev. 11:18 and 11:30. See also Chulin 63a.
See also Rabbeinu Bachya Lev. 11:2 and Lev. 11:43 as well as the Malbim on Lev. 11:29.
So the Torah would easily consider bats to be birds for a number of reasons: 1. They both fly, and therefore can be considered to be under the same general category of winged-flying creatures. 2. They would have similar names for this reason. 3. We're not necessarily expert in the differences between animals and their names in the Torah, so it might well be that we're misunderstanding to which creature the name actually refers.