I think it is a necessary test of self-awareness. In order to make the connection that "I am the being in the mirror" and that every action I do will be reflected in the mirror (or, vice versa, every action I see the being in the mirror take is a reflection of my own action), one must first have a concept of self. One requires a constant present-tense (at least) story of oneself: "I am doing this. Now I am doing that. Oh look, I am doing this and that guy is doing the exact same thing." And then there's the ultimate realization, "Ah ha! That's me!". That realization would make no sense if the individual has no sense of "me"!
[speculation] On the other hand, perhaps it is too strict. It seems that it would unfairly favor social species, since it requires not only recognizing another member of the species (and being able to associate the anatomy of another member of the species with one's own anatomy), but it may also require the recognition of intent. That latter requirement is complicated by differentiating between intent and compulsion: at some point, one must realize that the being in the mirror isn't trying to copy but rather it always, inevitibly reflects one's motions. So, perhaps a cat is self-aware, in a sense that it has a constant story in its head of what it is doing and what it intends to do, but it cannot easily ascribe intent to other members of the species and thus cannot figure out that the cat in the mirror is automatically mirroring all of its movements. [/speculation]
Herein lies the rub: there is no perfect test for self-awareness; we can cast doubt on any outcome. The best would be if these animals could communicate with us by language. In lieu of that, we have to think of such tests that demonstrate with reasonable confidence that the species carries all of the characteristics that we attribute to self-awareness. I think another good test would be determining whether the species can plan individual actions in the future; in order to say, "first I will do this, then that; finally, that will allow me to do that", it seems like it would reasonably require an understanding of self and the consequences of one's actions. An example would be birds solving sequential puzzles.