Dear Terrah McCracken,
Good day! Thank you for posting to Microsoft Community.
I understand your query here and you are absolutely right, in classic outlook you can move and pin a shared mailbox folders in the favorite folders section, but in the new outlook this feature is not yet available for shared mailboxes. However, this is a known issue that our product development team actively working on a fix. In the meantime, the feature adding and accessing shared mailbox as a primary account in the new Outlook for Windows is under active development by the team and it will start to roll out to all user in July 2024. Once this feature starts rolling out to all users, the Shared Mailbox will appear at the Account Level in the Folder list, and you can access shared mailbox features as same as your primary mailbox. Therefore, this feature will fix the issue related to conversation view, move/Pin shared mailbox folders in favorite folders section and other shared mailbox related issues. So that, I suggest keeping an eye on the Microsoft 365 Roadmap | Microsoft 365 and the Microsoft Outlook blog Things to look forward to in the new Outlook for Windows - Microsoft Community Hub for any updates on this feature.

I will also update here once the feature starts roll out on my side. Appreciate your patience and understanding.
Thank you for your precious time. Have a good day!
Sincerely,
Libeamlak | Microsoft Community Moderator