As that variant increases, the newer the aircraft, or larger the aircraft. I KNOW IT'S NOT LIKE THAT FOR EVERY ONE! But if anyone knew that, I thought it'd be the controllers...
I have been a controller for over 7 years. Most controllers could care less how many seats are in a plane. We are trained on performance. There is no reason why I need to know what the difference from a b737 to b738 to b739. They all fly the same to me. As an approach controller they all fly headings I give them and slow to speed I assign. Now knowing the weight class is very important for wake turb separation. That is something we are trained on. But it is impossible to know every single aircraft.
In the center environment climb performance and Mach numbers are important. The aircraft fly a little different but not by much. It all depends on the airline that flys the plane.
Most of the time if I ask a question to a pilot it is because we are talking about it in the control room.
BUT, if the controller doesn't know the basic type of aircraft, that can be dangerous as well. A couple of years ago I was on with Bakersfield Approach (where I assume the radar shows tags of speed and altitude just like everywhere else) and was descending through 11,000 doing 320 knots groundspeed. The controller gave another aircraft a traffic call about us saying "Cessna ……., you have traffic at your 11 o'clock, descending out of 11, a Bell Helicopter" He was not familiar with our filing code of B462 and just assumed we were a helicopter, not a 4 engine jet which we really were. So now the Cessna is scanning the skies for a whirly bird. He could of at least asked us what we were, as we get most of the time from other controllers.