So you would need a plane for each character that you want selectable. Then you just swap textures (or UV offset if you are using an atlas with smaller character images). Then each plane would get a collider on it.
You can keep the state of each button in a manager, or probably playerprefs or an XML or wherever you want and you can either name the planes the name of the character, or give them a tag or something, but when you click the screen, in the Update function do a raycast and see if it hits something. Then check the gameobject name or tag (whatever you set) of the hit object and you'll know what they clicked on. Then from there you can see if the plane that was clicked on is an available character or not.
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