I had a seeded run (Save1) going in my main profile and I wanted to test something, so I wanted to use my second profile. This was the first time I loaded it after the new demo came out.
I loaded my second profile, then used "Unlock all". After that I went to start a test run (Save2), but noticed that I had the run (Save1) from my previous profile loaded in "Continue". I thought that was weird, then I started a new run (Save2), and quit to menu. WITHOUT clicking "Play", I loaded my profile 1, clicked play, which opened Continue menu for Save1.
I closed that menu, switched profile1 to profile2, now I clicked "Play" on profile2, and I noticed that the Save2 was not there in "Continue", and instead Save1 was there.
It is hard to explain, but MOST LIKELY it's the same bug as #1204808681322192956 https://discord.com/channels/1116389027176787968/1205188819704483901, but cross-profile.
Also after recreating ^ multiple times, I created a fresh profile and tested it with it as well, it also worked. But you needed to "load" a run (look at it from "Play" in main menu) in the new profile first.
TD;DR
So if you start a run on profile1, quit, click "Play" to "look" at the run, it will become "cached", then you switch profiles, and when you click "Play" the "cached" save will be loaded in instead. The requirement is that the other profile has a run going (I think)