It seems that this issue remains unresolved. While I'm currently running Arch Linux, this issue occurred with both of my previous Linux installs (Ubuntu-MATE 21.10 and KDE Neon 5.23). On Windows 10 and 11 when I had them installed, this issue never surfaced.

I have done what the official help page recommended for Arch (installing gnome-keyring and starting the daemon), and that hasn't resolved the issue. Granted it only mentions a token failure rather than this exact issue, but I wanted to cover all my bases. This isn't an issue that makes playing impossible, but it is rather annoying to have to go in and log in every time that I start my computer. If Mojang actually has a fix for this, or if anyone who has had the same problem has a fix themselves, I'd greatly appreciate it.