In order for the tame to work, the dead animal MUST have been killed by you or your tame. If you don’t see the yellow text, it didn’t count and they won’t eat it. Saying this because I killed a golem with my thylas bleed attack, which does not credit me with the kill.
In order to tame you have to let it follow you while you kill creatures. It will then proceed to eat the corpse, the taming bar raises with each bite it takes. Depending on what you kill determines how fast the taming bar rises. Phiomas worked well because it takes several bites. Keep your eye on it so it doesn't fly off and be careful not let it take damage or you will have to start over again.