I had a similar experience - I was merrily reading along with the “kindle cloudreader” in Brave, then accidentally hit the “previous page” button for the browser instead of the previous page of the book.
When I went back to use the cloudreader, it said it had to be installed but that my browser wasn’t supported. I didn’t even change Brave versions or anything.
It’s almost as if a leftover instance of the cloudreader were still running from Chrome, then I killed it, and amazon detected the browser when I tried to re-open it.
This was apparently not the case, however, since I re-started the cloudreader in Chrome, but still couldn’t run it with Brave.
The cloudreader would be handy, but for now, the program from here works fine: