New message notification in shared mailbox

It is normal. Have you setup the shared mailbox as a secondary Exchange account?

https://www.msoutlook.info/question/new-mail-alert-for-delegate-or-shared-mailboxes

There are workarounds in the link above.

Thank you Jackie -  
I gave multiple users full permission to this mailbox in EAC and the shared mailbox automatically added to the users outlook as additional mailbox. This is a cloud mailbox in a hybrid deployment. The users with on-prem  mailbox can see the number of unread messages while the users who has cloud mailbox (o365) don't. They would have to manually click "update folder" in send/receive to view the number of unread messages.

the shared mailbox automatically added to the users outlook as additional mailbox.

Remove the shared mailbox and add it as a secondary Exchange account manually.

https://www.experts-exchange.com/questions/28613764/Is-there-a-way-to-get-Desktop-Alerts-for-a-Shared-Mailbox-in-Office-365.html

Or try the method in the link above.

FYI - I have already look at the link you provided above. I added the Inbox Favorites to no avail.   I have a lot of other share mailboxes that don't have this problem.