Hello,
Thank you for posting in Q&A forum.
Categories should be preset by users who have full access to the mailbox, in which case you are the administrator of the shared mailbox, and once you create the category, other users of the shared mailbox will also be able to use them.
You mentioned that "the categories are not syncing properly" and I suggest you check your permissions. If you're already an administrator, you can remove everything from mailbox delegation for shared mailboxes in the Exchange admin center, and then re-add them to check the results
I hope the information above is helpful.
If you have any questions or concerns, please feel free to let us know.
Best regards,
Jill Zhou
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