Looking at the full diff[0] it certainly looks like it's using ~/.cache (and has been for some time), but I cannot see anything about ~/.local/share, no.
[0] https://hg-edge.mozilla.org/integration/autoland/rev/8a6d6c0...
[0] https://hg-edge.mozilla.org/integration/autoland/rev/8a6d6c0...
Which already is a huge improvement and better than bikeshedding for decades that they also should use $HOME/.local/share/mozilla in addition.
Either they adopt XDG fully, putting cache files where they belong, or don’t just change things haphazardly for little benefit.
You can exclude $HOME/.config/mozilla from your back up all the same anyway if that causes you some issues.
I personally appreciate them not cluttering $HOME with this move. It is better than waiting another 21 years for them to support XDG spec fully by splitting share and config.
And then just not have Firefox backups?