It has been discovered this is an upstream bug related to Resist Fingerprinting. Before attempting to sign in to Firefox Sync, go to about:config and change privacy.resistFingerprinting to FALSE. Then sign in and set up syncing.
Once done, I've re-enabled RFP and syncing still works, for now.
Source: Firefox Sync doesn't seem to work (#310) · Issues · LibreWolf / Browser / Windows · GitLab