Hello, @Iniobong Nkanga,
Welcome to the Microsoft Q&A platform!
Based on your description, I understand that you want to add a specific user as delegate on shared mailbox but fail to find him in the list in EAC. After testing, I found that the users selectable in the Delegation list are the same as the Mailbox list. Therefore, please check that you have added the target user to the EAC before adding him as a Delegation.
If you still cannot find the target user in the list, you can check the mailbox successfully added by using Get-Mailbox command in PowerShell. As below, you can find the mailbox information you added in PowerShell.
Also, you can use the Add-MailboxPermission command to set the target user as the delegate for a shared mailbox directly in PowerShell. For more detailed information and commands, you can refer to this document: Add-MailboxPermission (ExchangePowerShell) | Microsoft Learn.
Please feel free to contact me if you have any queries. If my reply is helpful to you, please mark it as the answer so that other users can refer to it. Thank you for your support and understanding.
Best,
Alex Zhang