The Milky Way has no categorising designation. It is just called "the Milky Way Galaxy". No NGC or IC or M or any other designation. Yes, it has galactic type of description, a barred spiral, of which evolutionary type I cannot recall. But it is just the plain old Milky Way Galaxy. That's it.
I guess it is a bit of a quirk of the way that Science, astronomy for that matter, has evolved itself. The ultimate discovery that the galaxies we know today as being 'island universes' is less than 100 years old. Previously they were thought to be 'spiral nebulae' and the such, but just different types of nebula. And as astronomy progressed, even though we came to understand that we ourselves are part of one such 'island universe', no one thought to name it otherwise as MW, and the name Milky Way just became the accepted name, and no other designation has been given. That it is called Milky Way is nothing more than the accepted scientific nomenclature of Greek and Latin origin.
Just 'one of those things',
