Q: Can I move my ally's unit to my own base with Unforgiven, the Yasuo legend? Does that mean it's mine now?
No, you can't do this. Any time you try to move something you don't control into your base, it moves to its controller's base instead. This is hard-coded in the rules, similar to how cards you own can ever be in your own hand, trash, or deck, even if something tries to put them in a different when. Similarly, you and your ally can never both have units at the same battlefield; the rules prevent any move that tries to make that happen.
A few cards, including Unforgiven, were printed with English text that suggests misleading outcomes in these situations (but were printed correctly in Chinese). Those cards are receiving errata to clarify what they can and can't do.