The Inbox — Assignment and status
Both are ordinary fields on the ticket, which is why they show up everywhere else too — in views, filters, reports and automations.
The Inbox uses a People field for the owner. It picks one named something like Assignee or Owner, or the only People field if there is just one. If the table has none, Admins get a one-click Add an Assignee field button — the Inbox never adds a column to your table on its own.
For status it uses the Select field the chat widget or the mailbox nominates, or one named Status. Changing it here is an ordinary edit: it is audited, it can trigger automations, and if the widget publishes that option to visitors, the customer sees the ticket move.