If an O365 user is requested to get access to another mailbox, so they can switch to that mailbox in O365 Web App, but they do not want it to show up in their Outlook, this can be done with PowerShell.

1. On your office PC, open PowerShell as admin

2. Install ExchangeOnlineManagement module -- this step only needs to be done once

Set-ExecutionPolicy RemoteSigned

          Choose A for Yes to All

Install-Module -Name ExchangeOnlineManagement

          Choose A for Yes to All

3. Connect to Exchange Online with an admin account

Connect-ExchangeOnline -UserPrincipalName <O365 Admin email address>

Example:
Connect-ExchangeOnline -UserPrincipalName admin@company.com

          You should get a Modern Auth popup dialog. Sign in as usual.

4. First, remove any existing access to the mailbox (to make sure it will not show in Outlook)

Remove-MailboxPermission -Identity <target mailbox email address> -User <user needing access email address> -AccessRights FullAccess -Confirm:$false

Example:
Remove-MailboxPermission -Identity mark@company.com -User spy@company.com -AccessRights FullAccess -Confirm:$false

         Note: This will fail if they did not already have the permissions. Fine to proceed.

5. Add Full Access (i.e. Read and Manage permissions) to the user while preventing AutoMapping (showing up in Outlook)

Add-MailboxPermission -Identity <target mailbox email address> -User <user needing access email address> -AccessRights FullAccess -AutoMapping $false

Example:
Add-MailboxPermission -Identity mark@company.com -User spy@company.com -AccessRights FullAccess -AutoMapping $false

6. (Optional) If they request to be able to “Send as” or “Forward from” the mailbox, they will need the below permission as well.

Add-RecipientPermission <target mailbox email address> -AccessRights SendAs -Trustee <user needing access email address>

Example:
Add-RecipientPermission mark@company.com -AccessRights SendAs -Trustee impostor@company.com