I am not sure but I thought this option was standard. When you pick up a task from the sprint board it will automatically assign yourself to this task because you pick it up. Is this option still there?
It might have been a bug (feature?) that was corrected in the move to the new navigation. I was able to replicate this in current TFS2015, but not in AzDO 2019 Server or Services.
It used to be when you were on the board view for the sprint and grouped by backlog, you had the option to filter to a specific team member. When you were filtered to that team member then any board adjustment would translate into a re-assignment to the filtered team member. I remember it was annoying because I would sometimes accidentally swap the ownership when I was changing the status.
Since you can group by PBI but not filter by user anymore, it doesn't look like that behavior is maintained in the new navigation.