That sounds like the mirror test, and similar tests of self-awareness. Which indicate that "sense of identity" is pretty abstract and high-level.

From wikipedia:

Animals that have been observed to pass the mirror test include:
All great apes:
Humans – Humans tend to fail the mirror test until they are about 18 months old, or in what psychoanalysts call the "mirror stage".
Bonobos
Chimpanzees
Orangutans
Gorillas – It was initially thought that gorillas were unable to pass the test, but there are now several well-documented reports of gorillas (such as Koko) passing the test.
Bottlenose dolphins
Orcas
Elephants
European Magpies

Of course, different tests are likely to give you different results, but this seems like a start.