Why does Principal Vagina say, "no relation" after introducing himself?

The joke is that the Principal is saying his name has "no relation" to the body part. He is constantly attempting to clarify that his last name is "not a joke" and that it's a real, "possibly Scandanavian" last name.

When people who have the same (or similar) name as someone famous introduce themselves, they will often get questions as to whether they're related. Think if an American was named Richard Clinton, or Travis Trump.

After a while they will anticipate these questions, and as part of their self-introduction just state flat-out that they aren't related to that famous person it sounds like they are related to. Just to avoid having to be asked: "Hi, my name is Richard Clinton. And before you ask, no, I'm not related to Bill." This can be shortened to "No relation."

This is what the principal is doing. Only instead of a famous person, it's a body part.

I always understood it as a reference to "not being a pussy". Thus he is saying his name is not a reference to his character.

It could also be a nod to Die Hard's Agent Johnson and Special Agent Johnson.

AJ: I'm Agent Johnson
AJ: And this is Special Agent Johnson
AJ: ... No relation

The joke there is they have the same surname but they're clearly of different ethnicity.

But in relation to Rick and Morty, Johnson is American slang for penis.

To me, having grown up with a TV in the 90's (for good reason: moved to USA for learning the language and culture), this phrase was an instant reference to Tiny Toon Adventures cartoon series, where two major characters - Buster and Babs Bunny (sometimes with Bugs Bunny in the loop) - always introduced themselves with a "No relation!" disclaimer.