you decide what you want to do, you want it, you make it!!
is there a way to make UI elements only interactive and visible to one play only?
Yes. You make your events do that!
In the UI, I will create a token that represents the player but when the other player gets created I don't want The Player 2 Token to Overlap the first one.
You don't want that to happen, so what do you want to happen? Do you want it destroyed? destroy it. Do you want it to move? Move it. Decide on what you want, and then make it happen