@Artem Shaturskyi , Thanks for posting in Q&A. Based as I know, to let contacts created in Outlook app to sync with native contacts app. We can create an app protection policy and an app configuration policy.
For app protection policy, we can set "Sync policy managed app data with native apps or add-ins" as allow.
For app configuration policy, we can set "Save Contacts" to Yes.
You can also set the synchronization of contact fields to the native app by the following setting.
For more detailed steps, you can refer to the following link.
Note: non-Microsoft link, just for the reference.
Hope the above information can help.
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