While not canon, Sonic Universe implies that Omega runs off of a Chaos Drive, and in IDW he’s able to still “function” as just a head. Both of these seem to suggest that Omega doesn’t have an animal inside him.
My guess is that after seeing Gamma commit robot suicide after realizing that he was basically just a tiny bird inside a giant mech suit, Eggman decided to stop using animals to power his more advanced/sentient robots to make sure that kind of thing didn’t happen again. Omega is the last of the E-series, so if any of them were to have some drastic design change like this, it’d be him.