No — only one person can reserve a given wish at a time. Reserving an item locks it, so a second person can't reserve the same gift until the first reservation is cancelled.
That's the whole reason reservations exist: to stop two friends from buying the same thing.
How the lock works
First to reserve wins. Once someone reserves a wish, anyone else who tries gets a message that it's already reserved by someone else, and can't reserve it. If that person changes their mind and cancels, the item becomes reservable again and someone else can claim it. See "How to cancel a reservation"
The lock is per item. It applies to that one wish, not the whole wishlist. Other items stay open to other gift-givers.
A wish has no quantity field. Even if the owner would happily accept two of something, each wish holds only one reservation. If you want to gift the same kind of thing as someone else, ask the owner to add a second wish for it.
Reserving still isn't buying. The lock prevents duplicate reservations on Wishpicks; it can't guarantee no one buys the item elsewhere, since the actual purchase happens at the store. See "How does gift reservation work?"