The first time you reply to a message sent to the shared mailbox, you need to add the shared address to the From drop-down box. Here's how to do that:
Open Outlook on the web and go to your shared mailbox.
Open a message that was sent to the shared address.
Click Reply.
At the top of the message, choose
> Show From.

Right-click the From address and choose Remove.

Type the shared address and send the message. The next time you reply from the shared mailbox, its address will appear as an option in the From drop-down list.