Today pilots and students get a "class rating" that certifies they know how to fly that general class of airplane such as single engine land. You might learn in a Cessna 150 or 152 and then switch to a 172 for the larger engine and more fuel so you take bigger cross-country legs. Those planes are in the same class so it's an easy transition.
Even so you can't just go rent a 172! You'll need someone to sign off saying you know how to fly the 172 or you'll have trouble renting the plane or possibly with insurance. Insurers want to see some time learning to fly a new model before going solo.
So if you learn to fly an electric airplane in the US you'll still get Private Pilot - Airplane, Single Engine Land (PP-ASEL) but you will require further training to hop in a gas engine plane that's also single engine land.
The "land" part is to distinguish those planes from sea planes (planes which can land on water) which are different enough for their own class.
Flight training goes through various phases. Just like traditional education, you have primary, elementary, and secondary. Primary flight training consists of the fundamental “stick and rudder” skills which all pilots from a Cessna to a 747 must posess. In fact, the US Air Force academy starts all fighter pilot candidates out flying gliders for this very reason. Electric flight will absolutely revolutionize this portion of training because the primary limiting factor in one’s ability to perfect those skills is cost. And the overwhelming majority of cost in a single engine aircraft is the engine itself; gas, oil, maintenance, overhauls. All of these things will be obliviated, freeing pilot’s resources to obtain multiple times the amount of hours as is currently the case.
Not only will this be good for pilots, but the industry and public safety as a whole will be greatly enriched by the increase in basic flight skills that have started to slip over the years due to the exorbitant cost of primary training.
I've only flown 12 hours in a traditional aircraft (piper warrior iii), but there's certainly a few things I'd imagine a pilot would want to know before taking off in something traditional (oil pressure, manifold, switching fuel takes, checking fuel quality, etc.). Maybe just enough training in a traditional aircraft for a few extra hours to be rated properly.
I'm certainly excited for the possibility for electric aircraft, hopefully in my lifetime. Less maintenance, less things to go wrong, cheaper, (quieter/less vibration?) etc.