Dear @Murad Ismayilov,
Welcome to Microsoft Q&A Forum and apologize for the inconvenience.
This behavior may sometimes occur under certain conditions related to server-side synchronization. After researching this behavior, we found that it aligns with an active service incident under the reference code EX1238375. This specific service behavior currently affects users attempting to access the "My Templates" and "Viva Insights" add-ins across multiple platforms, including the Outlook desktop application, Outlook mobile, and Outlook on the web. It seems the automated system may require a little bit of time to process these requests as the service is being restored. Please refer the following article: My Templates are missing from Outlook
Therefore, in this case, please try to reach out to your organization's IT Administrator and ask them to submit a formal support ticket through the Microsoft 365 Admin Center
In the meantime, you can use Signatures as a temporary workaround, I know it's not the same, but it does have the similar function to My Template.
I hope this clarification helps align expectations and provides you some insights in this concern. Wish you a pleasant day!
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