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.
A program a watched a long time ago showed researchers working with dolphins. They would draw something with a marker on the side of the dolphin, and then it would race to an underwater mirror and contort its body to view the mark. The researchers said that this demonstrated self-awareness. As for being compared to humans, meh. I donno.
Usually in this context, "self-aware" is usually used to indicate the ability to recognize one's own image, voice, smell, etc. The research referred to is mostly several experiments of showing dolphins to a mirror and determining that they can recognize a) That they are looking at a reflection, not a window, and b) They are able to distinguish themselves from other dolphins.
This only applies to one (quite narrow and literal) definition of self-awareness, mind you, but it's something you only see in apes and dolphins.
You also need scientific experiments in order to argue that humans are self-aware. Indeed, young humans aren't self-aware, hence Piaget's classic experiments on development to show the emegence of self-awareness.
The article does allude to a couple of experiments in the article ("pass" option if they can't answer a question, self-recognition in a mirror). I agree it needs to be studied in more depth to make these assumptions and certainly to confer new rights.