For Azure Services, Microsoft Wire Server will integrate with one of these recommended zone names, and unfortunately there is no "custom domain" scenario in this case. This answer provides additional info of what is going on.
From the docs:
When you create a private endpoint, the DNS CNAME resource record for
the storage account is updated to an alias in a subdomain with the
prefix privatelink. By default, we also create a private DNS zone,
corresponding to the privatelink subdomain, with the DNS A resource
records for the private endpoints.
When you resolve the storage endpoint URL from outside the VNet with
the private endpoint, it resolves to the public endpoint of the
storage service. When resolved from the VNet hosting the private
endpoint, the storage endpoint URL resolves to the private endpoint's
IP address.
Workaround: The only workaround is an internal Application Gateway. I've never done it for Blobs, but for Web Apps you must rewrite the host HTTP header, otherwise it won't work. Microsoft.Web services require that header to match the service, which must be the public name, as explained above.
There is another workaround (which I won't recommend because of this issue), which is validating your private domain name in the public DNS with a TXT record which you can later remove.