Leagues have set transfer rules - i.e windows and registrations forms, so players are "league tied" until a transfer happens. Cups are easier than leagues, you can only play for one team in that cup for that year, You don't need a transfer process.
It can work both ways, in village cricket, one year my Dad's team had been knocked out of a cup 1st round and he hadn't played that game, so, cleared by the cup organiser, he played for another village as they were short of players. I cant see the FA, EFL etc allowing us or anyone else to borrow a non cup tied payer for a game or two though.....