No, the status field is global across all products and you will not be able to generally configure different sets of statuses for different groups.
A suggestion would be to add different custom statuses in Admin -> Workflows -> Ticket Fields and click on status to add the options to the dropdown list.
Once this is done, please navigate to Admin -> Workflows -> Automations -> Ticket Creation -> Add new rule. Once there, please add a condition that says when the ticket is assigned to a particular group - then the action would be to set status as this "custom option."