How are tamed animals treated in terms of abilitys?

In a campaign I am playing in the dm allows the taming of wild animals, but you have to devote time and resources to training them to follow commands after the initial taming. Would these animals after the training just have the basic stats for that creature, or would they get the same special abilitys over time as a beast master rangers or druids animal companion? For example in my current situations I tamed an owl, so would it just use the owl statblock or would it get the abilitys of a beast master ranger or druids animal companion of being able to take class levels and all the other stuff that goes with it?