Hi Sergey Avramenko,
I'm glad that you were able to resolve your issue and thank you for posting your solution so that others experiencing the same thing can easily reference this! Since the Microsoft Q&A community has a policy that "The question author cannot accept their own answer. They can only accept answers by others ", I'll repost your solution in case you'd like to "Accept " the answer.
Issue:
Can't enable Microsoft Entra authentication for SQL Server 2022 enabled by Azure Arc
Solution:
Issue resolved by @Sergey Avramenko
Service-managed certificate, which created one in Key Vault but showed a message about needing admin consent for the app 'AZURE-APP'. After setup, login via SSMS failed, and the Admin Login in Entra ID wasn’t configured. Once I set it up again using the Azure-generated certificate, the settings saved successfully, and I was able to log in using Entra MFA.
If you have any other questions or are still running into more issues, please let me know. Thank you again for your time and patience throughout this issue.
Please remember to "Accept Answer" if any answer/reply helped, so that others in the community facing similar issues can easily find the solution.