Problems with iMessage in Big Sur not auto-syncing my messages when signed into iCloud account. I have a MacBook Pro 13" for work, a personal MacBook Air 13", and an iPhone 11 all using the same iCloud account.

Just to verify:

You have text message forwarding on your phone set to both devices? (Settings > Messages > Text Message Forwarding).

You have messages in iCloud enabled to sync (Settings > {Your Name} > iCloud toggle on messages) on phone

on Mac, open messages. Messages > preferences > iMessage enable

I've only experienced the issue you describe once. Signing out and back in fixed it for me.

I do not have a togga key board,and my mac book air i had to sell,verizon put me in covid ,were all dissconnceted divices were and my wifi thay found out.So i got so manny linkens in my mac book air i could not even work on it .So i cleaned it out and sold it mac boone