Why is 2 considered a prime number?

This is really a question of terminology. The current notion of an integer that is unrepresentable by a product of other integers is given the name "prime number," and you're asking why the term "prime number" doesn't refer to some other set of numbers.

Ultimately, it comes down to what is useful to mathematicians. Terminology isn't invented simply to pass the time; terminology is invented because somebody observes a concept repeated and then decides that that concept is common enough that it is worthwhile to give it a name. The current notion of prime numbers came up more often1 than the notion of "prime numbers excluding 2," which is why the former was given a short name and the latter made more ugly-looking.

The redefinition you're suggesting has occurred before—historically some mathematicians included 1 in lists of prime numbers, but it led to inconsistencies and required frequent usage of "prime numbers excluding 1," so eventually it became common not to include it. I like the quote given by MathWorld's article regarding this issue:

2 pays its way [as a prime] on balance; 1 doesn't.

How can two be divisible by a number other than itself and one?

You are right that it obviously can't, but it's fine to make arbitrary statements about the elements of the null set. You can say that every element of the null set is even and odd and counterexample to every unsolved problem in mathematics, because there is no element. Disallowing that kind of reasoning would be the same kind of inconsistency that happens when you exclude 2 from the prime numbers—you invent a special case that need not exist. Special cases being elegantly extrapolated from the common pattern is something to be celebrated, not eliminated.

1 Famous examples include Goldbach's Conjecture and the Fundamental Theorem of Arithmetic, both of which rely on the inclusion of 2 as a prime number to work.