When I launch Chrome on Linux Mint, it asks for being the default browser. I confirm and it is remembered between browser relaunches. But when I shut my laptop down and then turn on again, default browser is not set anymore. Also, after I press Set Default in Chrome I can see that default browser in Preferred Applications is still unset. But if I set it there it is also persisted until next system restart only.
Haven't used Linux Mint for years, but can you verify that
~/.local/share/applications/mimeapps.list exists? There's a possibility that if this file doesn't exist then chrome will fail to set itself as the default browser.
If it doesn't exist then you could do something like:
mkdir -p ~/.local/share/applications # use -p to create intermediate directories as required touch ~/.local/share/applications/mimeapps.list # then create the file
Then try to set default browser again and see if that works.