Yeah, it looks like a pretty modern bathroom and tub. It certainly has an overflow drain.

Edit: Two other notes. 1) The age is irrelevant, even some Victorian-era claw foot tub would have an overflow. 2) I can imagine a scenario where the drain itself is hairclogged at the tailpiece such that it is draining slower than the tub is filling, thus potential overflow.

Edit: From Sharon's video, a trip-level stopper/overflow plate: https://youtu.be/CKuTdZ5T8Ao?t=138