We'll always have the Matrix problem where it's hard to "prove" self-awareness in others (Descartes does a pretty shitty job), but inasmuch as we have reason to assume other humans have self-awareness, most scientists feel comfortable making the same assumption for animals that pass the mirror test and do well on metacognitive tests where they grade their own mental processes. Examples of the latter include the "pass" test described in the article as well as tests Herbert Terrace conducted in which monkeys were asked to perform various tasks and, each time, bet a number of M&Ms proportional to how confident they were in their solution. Sure enough, they tended to bet a lot on answers they got right and much less on ones they got wrong.
Note that such tests, especially the mirror test, aren't bijective. If you pass the mirror test, we can be reasonably sure you're self-aware. However, if you don't pass the mirror test (which I didn't as a toddler), that doesn't mean you can't be self-aware.