Is there biological evidence for self-awareness in animals?
Yes, there is a lot of evidence for self-awareness in animals. The gold standard test for self-awareness is the mirror self-recognition test (MSR)--we can tell from the behavior of an animal exposed to a mirror whether or not it can tell itself apart from other animals of the same species:


  Animals that possess MSR typically progress through four stages of behavior when facing a mirror: (i) social responses, (ii) physical inspection (e.g., looking behind the mirror), (iii) repetitive mirror-testing behavior, and (iv) realization of seeing themselves. (Plotnik et al. 2006)


That quote was taken from the abstract of a paper showing self-recognition in a single African elephant, but there is evidence for self-awareness in magpies (Prior et al. 2008), dolphins (Harley 2013), chimpanzees (Povinelli et al. 2006) and orangutans (Suarez and Gallup 1981). Many of these studies use the Gallup mark test (Wikipedia):


  Gallup anaesthetised the chimpanzees and then painted a red alcohol-soluble dye on the eyebrow ridge and on the top half of the opposite ear. When the dye dried, it had virtually no olfactory or tactile cues. Gallup then returned the chimpanzees to the cage (with the mirror removed) and allowed them to regain full consciousness. He then recorded the frequency which the chimpanzees spontaneously touched the marked areas of skin. After 30 minutes, the mirror was re-introduced into the room and the frequency of touching the marked areas again determined. The frequency of touching increased to 4-10 with the mirror present compared to only 1 when the mirror had been removed. The chimpanzees sometimes inspected their fingers visually or olfactorily after touching the marks.


Basically, an animal which has seen and recognized itself in a mirror recognizes the mark as being new, so it touches its own body on the marked area, which shows that the animal recognizes the image in the mirror as itself.