I was in the same boat and after some Googling I found that you need to have "Displays have separate Spaces" checked in the Mission Control preferences. For some reason it requires you to log-out and back in after activating it, but afterwards I can indeed use split view as described.

I don't remember changing this setting, but it's always possible I did years ago. I have never been a big user of Spaces and somehow missed out knowing Split View was ever a thing, but it's actually pretty neat.