In order to have self-awareness, you need to have semiotic mediation between experience and the environment, to be able to recognise the self as an object and be able to mirror that object in the reflected world. You also need to have some kind of dialogic cognition, so that you can understand that there are alternative perspectives that do not centre you in experience. You need to be able to construct an identity in order to provide that self with continuity over time. I am extremely unconvinced that almost any animal has these abilities, and would not be at all surprised to discover humans are unique in having them all (or, indeed, any of them).
Certainly, the most likely contenders (like chimpanzees and apes) do not have these abilities. For example, a chimp can pick up and move a chair and then pick up a stick in order to get down some bananas. However, if either chair or stick is outside their visual field, they are unable to do this task, even if they have previously done it. This is because they have no internalised representation of their visual field. Without the ability to semiotically mediate the world, it is ephemeral. Without the ability to hold a narrative, there is no self-continuity and so no self.
None of which is to say that animals can’t suffer. The complement of an objective self is subjective experience. If animals cannot self-reflect then all they have is subjectivity. Suffering is the subjective experience of not having your needs met, which is fundamental to being an animal. And without self-continuity, there is no future or past, only the now. As such, suffering is absolute, because there is no conception of experience without suffering.