I think UEFA keep the trophy, the club that wins it have to give it back for marketing, promo, etc. They get a replica if they've won it 5 times.
The winners names are engraved on the main part at the back of the trophy, but I didn't get a proper look around the back on the day. Had a good look at the UEFA Cup, that looked like the real thing, plenty of dents and scrapes, a replica would be spotless I imagine?!?