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Hi @Mette Viborg
The error in the Outlook log shows:
OwaUserHasNoMailboxAndNoLicenseAssignedException
This indicates that Outlook was able to locate and authenticate the account, but it could not find an Exchange Online mailbox associated with the user, or the user does not currently have an Exchange Online license assigned. To resolve this, assign a valid license to the user.
You mentioned that when signing in with @ANUIA.onmicrosoft.com, it shows the account does not exist. This could indicate that the User Principal Name (UPN) was changed previously from the .onmicrosoft.com address to the custom domain address.
Could you please confirm whether the UPN for this admin account was changed at any point?
As a test, please try signing in to the Microsoft 365 Admin Center using the @anuia.dk account at:
Microsoft 365 Admin Center
If you can successfully access the Admin Center, please also check the user's current sign-in name (UPN) and whether an Exchange Online license is assigned to the account.
Please try and let me know how it goes.