I do recognise that a lot of it is probably just playing up for the joke, but I assume it has to come from somewhere. I regularly see posts where basically Person A sends an odd image/message, then follows up with "wrong number, sorry" and Person B responds with "wait stay".
Do Americans, as a rule, not save phone numbers? Or is it purely a bit that has become a trope?
It's a new trick from scammers. They engage you that way and try to get money or gift cards from you.
(Bad) Reddit has a complete subreddit about it and people post how they string them along etc … it is hilarious.