Gift Shop tracks which modes have been chosen, not who chose them. If another player gains control of Gift Shop and then visits it, they won’t be able to choose a mode you chose while you controlled it.
2022-10-07 • wotc
If there are no more modes to choose, the Gift Shop is sold out. The ability is removed from the stack with no effect.
2022-10-07 • wotc
If you choose the last mode and don’t put a sticker on anything, you still chose that mode. It can’t be chosen again.
2022-10-07 • wotc