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.
Sometimes permissions can be a problem: if one of the settings files under your home directory has ended up owned by root, then the session cannot save when it shuts down.
find . -user root, run from your home directory can find problems like this.