Turns out there are two kinds of Microsoft Office 365 Shared Mailbox:
The one under Groups in 365 admin center, called Microsoft 365 group;
And the one called Shared Mailbox in exchange admin center.
To create rules for within Shared Mailbox, it has to not be the 365 Group, but the Exchange Shared Mailbox, which is free under 50GB.
Interesting how we can create rules to trigger flag change as well as category assignment. Flag is to indicate something like a reminder to follow up in set amount of time; while category is like a label. I could be wrong, but that's not the point of this entry.
To create rules to mark an email with flag, that has to be done in the Outlook client; while the rule for the OWA shared mailbox can do categories but not flag. To get to this shared Outlook account, simply go to the profile (right of the settings) in the outlook page > select "Open another mailbox", and type in the shared mailbox name.
However, to do this in the Outlook app/client, one must make sure that shared mailboxes don't automap to the user's account, meaning that the shared box would appear automatically in the user's Outlook upon sharing. Or secondly, to make sure that the shared mailbox is not listed in the user's account settings under advanced. For the first one, the solution sounds easy, basically just run the powershell script that removes automapping. It can be added back again.
I've also learned that we can turn Microsoft 365 Group to have like distribution list feature by delegating "Send As" to members in Exchange admin > recipient. And by going to the Group's settings in 365 admin to check "let people outside the organization email this team", and check "send copies of team emails and events to team members' inboxes".