Yeah, that's how it was supposed to be, the PvE is "Overwatch 2", the PvP is "Overwatch" with some unclear thing about what it is you get F2P and what you get from owning the original. I think it's just skins and crap like that because they backed off PvP being restricted only to the first game. Maybe playlists or something. The long term goal is to transition everyone to F2P Overwatch 2 so I'm sure at some point it'll just be a badge on your account and maybe some small collection of cosmetics for people who bought the original.
It's really what Valve did with Team Fortress 2, they just didn't call their PvE mode Team Fortress 3.