Yeah, we have several in my city. It used to be, that those were full of book crossing stickers, but I think that pretty much died out and people just put books in there without any (manual) tracking codes ;)
In the UK many smaller hotels have a public bookshelf where guests can swap their holiday reading
Is it public in a way that even those not staying in the hotel can leave and/or get books from it?
Not really - the bookshelf might be in the guest lounge or similar - typically a room with a seating area adjacent to the bar. The last place I saw one the bookshelf was on the stairwell to the rooms, where non-residents wouldn't be expected to go.
In France too, it's super common to find public bookcases in parks.