Once you enable the Round Robin feature within your Shared Inboxes, the members in the shared inbox will be assigned messages that are distributed evenly.
Let's say you have a Shared Inbox with 3 users
The first message sent to the Shared Inbox will be assigned to user 1 on the list and the next messages will be assigned to members in chronological order after that. The next incoming conversation will be assigned to user 2
and the next new message will be assigned to user 3.
To enable Round Robin for your Shared inboxes, go to Settings > Inboxes, select the inbox you want to enable this feature for and toggle the feature On.
While you can control which members have access to the Shared Inbox, every user who does have access will be in the Round Robin assignment rotation. Including the Organization Owner.