Hi there, syncing mailboxes between servers is not possible, not. the mailbox can only exist in one database at any one time , mounted on one server.
You can sync databases however using database availability groups, but that isnt the same thing of course.
https://learn.microsoft.com/en-us/exchange/high-availability/manage-ha/manage-dags?view=exchserver-2019#:~:text=A%20database%20availability%20group%20(DAG,high%20availability%20and%20site%20resilience.