Using Outlook for iOS for personal email and calendar management
We were able to resolve the issue.
The user once again re-installed the Outlook app from the App Store and when it asked them if they wanted to allow notifications, they accepted and then from there they were somehow able to add the shared mailbox.
However, they were unable to receive notifications for this shared mailbox and was only receiving them for their personal inbox. This was resolved by turning on calendar notifications for the shared mailbox.