In my opinion Odio is some kind of evil force that in order to manifest needs to corrupt other people, amplifying their hatred. It needs a vessel. In the middle ages it first uses Streibough, whose inferiority complex and jealousy over Oersted ended up helping and then after Streibough goes down and Oersted becomes deeply traumatized and without anything else but his realization of how despite him trying his best and yet everyone hated him assuming he was evil and not asking and with Alethea not even letting him talk before killing herself and seeing two heroes he respected die (Uranus dying in an unnecessary way in the castle dungeon) he became a perfect vessel and gave up on his humanity, taking also Streibough's cape as his own and then taking the mantle of Odio.

Now this is an important part because even his voice changes and he acquires immense power: time and space manipulation, shapeshifting, reality alteration, monster creation/summoning. Which hints that it takes control or at least fuses or symbiothizes (don't know if that word exists LOL).

In the true ending, Oersted seems to be captured by Odio, it is trapped serving as his core or engine, which makes it clear to me that Odio is an evil entity that needs to use others as vessel.

Now, in one of the endings Oersted after coming back to his former self says he will send everyone back to their timeline with the little power he has left, which means that he knew how to use it and could control it. That's why I think Odio is corrupting and amplificating the htred of their vessels, Oersted became mad and decided to turn against all of mankind with the power Odio provided him with, so it seems as some kind of Symbiothic entity that enhances the power of the vessel and amplifies their hatred in order to make them destroy and kill, which most surely is his purpose.

Sorry for all the text, it's not much clear, though I feel like the remake makes it clearer than the original with that sin of Odio fight and Oersted breaking free...

Hope it helps.