Hello @Akhil PB ,
Welcome to Microsoft Q&A Platform. Thank you for reaching out & hope you are doing well.
The error message "The client with object id does not have authorization to perform action" is related to permissions.
Could you please confirm if the object ID "eccbe23e-ce10-49ee-a742-9f8612377ef1" has read/write permissions over the Private DNS zone "privatelink.mysql.database.azure.com"?
Refer: https://learn.microsoft.com/en-us/azure/dns/dns-protect-private-zones-recordsets
Please make sure that the above object ID has Private DNS zones Contributor role on the private DNS zone "privatelink.mysql.database.azure.com", providing the required read/write permissions.
And if you would like to link a Virtual network to this Private DNS zone, then make sure that the above object ID has both Private DNS zones Contributor role on the Private DNS zone and Network Contributor role on the virtual network.
Kindly let us know if the above helps or you need further assistance on this issue.
Please "Accept the answer" if the information helped you. This will help us and others in the community as well.