CalendarBridge cannot support Microsoft 365 Personal and Family plans.
If you have connected a Microsoft 365 personal or family account to CalendarBridge you will see a “Calendar” in the list of calendars — but that calendar cannot be synced to outlook on your desktop and that calendar cannot send or receive invites via your email address. (Yes, it’s very confusing and we are not sure why Microsoft does it this way). The end result is that it looks like CalendarBridge is not syncing events, but in fact it is syncing events to and from the calendar that is only accessible in Outlook Web.
Do you have two Microsoft accounts with the same email address?
Another confusing thing that Microsoft does is it allows you to have separate “personal” and “work or school” accounts under the same email address – And many people accidentally create a “personal” Microsoft account under their work email address without even realizing it.
One way to tell if you have two Microsoft accounts is you may see something like this when logging into Microsoft:
If you are intending to add a Microsoft account setup by your work or school, make sure that you are selecting a “work or school” account during the login with Microsoft process. On the enter password screen, a login.live.com URL means you are logging into a personal account, whereas a login.microsoftonline.com URL means you are logging into a work or school account.

Microsoft work or school account login

Microsoft personal account login