In order to conceive of anything, including yourself, you need some way of creating and using symbols. There are many animals that understand various symbols. Dogs are pretty good at understanding human commands, and they can, to some degree, communicate to humans. Many other animals can interact with humans on a symbolic level, albeit a very basic level.
As far as I know, cats do not use symbols. Many cat owners think their pets are quite smart, but cats basically ignore anything we attempt to communicate to them. They do what they do when they decide to do it. Clearly, cats are not very intelligent.
I think if you can use symbols to pass information from one being to another (oh all right, I’ll let cats back in), then you can tell the difference between you and not-you. You can begin to understand you are not the center of the world. There are others.
My guess is that many animals do have a concept of self, however rudimentary it may be. I think that we would not feel as close as we do to so many animals if we did not see clear evidence that they are self-aware.
What that awareness means, I don’t know. It’s clear that self-awareness does not necessarily endow animals with reasoning capabilities. Self-awareness does not make them very intelligent. They have barely enough awareness to be able to communicate only the most rudimentary things. So I don’t think they contemplate existence or have moral codes. They do, I believe, have emotions.