I’ve seen these in a couple of movies (including the fully restored version of Metropolis The “paternoster” scene is in the apartment building, and The Tall Man gets off just as Freder is getting on, so they don’t see each other.
I don’t know if the etymology has been fully researched, but the story I’d heard is that it was called a “paternoster” because riders would be so afraid of some accident happening that they’d say the Our Father. (If the name was based on the rosary, I’d expect it to be called a “Hail Many”, or its equivalent in some other language, since the contiguous beads are for repeated sayings of that prayer.