It's possible, I have a lawnmower that works like that but I'd wager that a lawnmower battery is considerably lighter than a car battery. In fact, quick Google says a Tesla P100D battery is 625kg. So unless you are Bruce Banner, Clark Kent or Steve Rogers, a battery swap is probably out, unless it can be automated, which would probably be near as expensive as buying a second car and swapping car instead. Also of note is that the battery weighs more than the original Rover Mini.
Not on Tesla. May be on Chevvy volt? 200 KG. That can be done using standard equipment. I was thinking of "charge loading points" where they load up your card with a new battery set and take old set for charging.