@Danylo Novohatskyi Thanks for posting in our Q&A.
For this issue, organization can't see the contacts in the device, so it will not manage this information.
https://learn.microsoft.com/en-us/mem/intune/user-help/what-info-can-your-company-see-when-you-enroll-your-device-in-intune
I have done a lot of research and I find someone managed contacts sync in Outlook. Please refer to the following link:
https://www.robinhobo.com/how-to-configure-automatic-contact-syncing-from-microsoft-outlook-to-the-native-contacts-app-including-contact-fields-filtering-on-ios-and-android-byod-devices-with-microsoft-endpoint-manager/
Note: Non-Microsoft link, just for the reference.
Thanks for understanding.
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