Well just because we are part of the universe doesn't mean that the entire universe is self-aware. Take a human for example. The only part of the human that is self aware is the brain. Without it functioning we would just be there. Our limbs and other organs are there to allow us to be able to interact with other objects.

Take away our limbs we can still intereact. Take away our brain and we no longer are able to.

In that sense, just because a piece of something is self-aware doesn't mean everything of the entirety is too.