Yet we are the only ones with a concept of self awareness.
Well, it's not easy to test without language, but no.
As far as we can test for it, there are many species that show various types of self-awareness; the most popular test for self-awareness is the mirror test, which tests whether the creature understands that a mirror reflection is of itself. Quite a number of disparate species can do this, including some birds - there's even (disputed) experimental evidence that some fish can do it.
Other tests involve whether a creature has a mental self-image, e.g. to allow mentally projecting itself into novel situations - again, quite a few species show this, including some birds, and rats. Rats, chimps and dolphins (and probably many other species as yet untested) have also been shown to be capable of metacognition - i.e. showing awareness of the limits of their knowledge in some situation and acting accordingly.
Even octopuses (octopi?) appear to have a degree of self-awareness and theory of mind (they often go together) which is unusual in non-social creatures - particularly molluscs!
It's interesting that chimps pass the mirror test, but macaques don't. This sort of high-level cognitive difference in ability between closely related species is not unusual.
So, many species show different sorts of self-awareness to varying degrees. What usually happens with human exceptionalists at this point is a form of begging the question, where they narrow the definition of self-awareness to a specifically human level, then triumphantly claim that only humans have
that level of self-awareness - "
Only humans have human self-awareness!". Well,
duh.