Ever since the macOS 13.2 update, both banner and alert notifications in Mail for some people (including myself!) are no longer appearing. The badge on the dock shows the correct unread count but the popup in the top right corner does not display for new messages.
I’ve tried everything from toggling Mail notifications on and off, to fiddling with Focus modes. What did eventually fix the problem was completely removing the affected mail accounts from Settings -> Internet Accounts, and then re-adding them. Toggling the Mail slider alone didn’t seem to work, it needed a full remove and re-add.
Ventura broke notifications in a few ways and various fixes work under differing circumstances. If you’ve tried everything else, then give the above a try and hopefully it fixes Mail notifications for you.