Tornadoes wouldn't be assigned names, but a "Tornado ID". It would be an identifier consisting a ordinal of tornado in that season and an intensity number.
The supercells they are attached to are already assigned a letter-number code that sometimes appears in radar software. This is not generally shared with the general public, however, as they don't really help humans much—mesoscale storms are ever-changing and storms can form, merge, and fall apart over the course of 15 minutes. You definitely couldn't assign individual vortices an identifier, since they can sometimes last less than 30 seconds.
Including the intensity of a tornado in an identifier is not possible. It cannot be known until NWS visits the affected areas and assesses the damage. Sometimes the intensity assessment will change as more data becomes available. And sometimes meteorologists disagree with the official NWS rating—there are people still out there arguing about whether the Joplin tornado was really an EF-5 or if it was just a high-end EF-4, and that tornado happened in 2011.
Hell, sometimes we don't even know whether a tornado is actually on the ground or not since "on the ground" and "50 feet above ground" look pretty much the same on radar. Weak tornadoes often don't have a condensation cone to make them actually visible, and of course, if it's night, you can't see them at all. The only way to know for sure is, again, for someone to go out and look at the damage. And you can't do that when the maybe-tornado is actually going on.