Dear Patryk C ,
Good day! Thank you for posting to Microsoft Community. We are happy to help you!
Shared mailboxes will be auto-mapped to the primary account as you now see in outlook. But, in-order to support some of the features like the one you are looking for, you have to add the shared mailbox as a secondary mailbox in your outlook profile. Go to Outlook> File> Add Account as below and add the shared mailbox as a secondary account. After adding the mailbox successfully, you can see the mailbox under Account settings. Make the shared mailbox as default there to set the default send address as needed.
If you have set the licensed shared mailbox as an additional mailbox in your Outlook profile and set it as the default account. As far as I know, to make it the default for outgoing emails, you need to select the "Always use default account when composing new messages" option via file > option > mail > under Send Mail.
In this way, no matter whether you select the default account before creating a new email, the "From" field in the new email will display the account by default.
Your patience and cooperation will be highly appreciated. Hope you all the best!
Sincerely,
Kerry Chen | Microsoft Community Moderator
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