Hi @Rob Banks ,
As far as I know, the command Get-User returns no mail-related properties for mailboxes or mail users. I have done a test in my lab, the same result with you, the Get-User command only return display name for me. If you want to confirm if the send on behalf permission is assigned successfully, you could run below command to do it:
Get-MailboxPermission -Identity onpremsharedmailbox@domian.com -User "user"
In addition, to check if this permission is working well, you could send mails in outlook via send on behalf, once the mails send successfully, it indicates that send on behalf permission is configured well.
If an Answer is helpful, please click "Accept Answer" and upvote it. If you have extra questions about this answer, please click "Comment".
Note: Please follow the steps in our documentation to enable e-mail notifications if you want to receive the related email notification for this thread.