Update (1/21/26): Multiple issues have been reported in this regard. I am posting a brief update on all of those issues.
A folder (like Inbox) of a shared mailbox pinned under Favorites lose its maibox name and display just as 'Inbox' - This issue has been fixed in Client Version 20260108025 (New Outlook > Settings > General > About Outlook > Client Version), which has been released to production.
A folder (like Inbox) of a shared mailbox with the above mentioned issue, also does not display the count of unread messages in it - This issue has now been fixed in Client Version 20260121xxx and will be deployed soon to production, so hopefully by Mon 1/26/26, it will be resolved for all users in production.
Users who have a folder in a shared mailbox/folder added as a Favorite are no longer able to drag and drop a message in it, as they see a 'block' sign. They can successfully drag and drop a message when choosing the same folder in related shared mailbox folder tree - This issue has also been fixed in Client Version 20260121xxx and will be deployed soon to production, so hopefully by Mon 1/26/26, it will be resolved for all users in production.
Hopefully this update will help all users who are seeing any or all of these issues working with Shared Mailboxes/Folders in new Outlook. Thanks for your patience while we work to release fixes for these issues in production.