When a game has been launched on one Microsoft account, and you attempt to open the game in a second account, it will report the second account does not own Minecraft: Java Edition.

It should be noted that it does not impede the ability to play. You can still hit "Play Demo" and the game will open and authorize normally without problem, provided you're logged in with the correct account.

Specific steps to reproduce;

1. Open the Launcher and log into any account.
2. Hit play to start the game.
3. Re-open (or switch to) the Launcher, and switch accounts.

Your Launcher should now instruct you to play in the Demo, or to buy the full game. Again, this does not impede playability, and users can still open the game normally.

This happens even after logging out of and restarting the Launcher entirely.

Edit: It's come to my attention that this may be a duplicate of MCL-19984; however, this also specifically states a potential cause.