That's not being a protected class (which is a statutory issue), or even being a Con-law suspect or quasi-suspect class, that's simply application of the 1st Amendment (incorporated by the 14th). Its a whole different, unrelated area of protection.
> The experience of the neo-Nazi is roughly the same, except the first time it happens they take the swastika out of their profile picture and from then on everything works peachy for them.
Which pretty much also works for many victims of religious discrimination, but even so religion is (and political affiliation, Nazi or otherwise, is not) a protected class under fair housing law. So, insofar as its a difference between "Nazis" and "blacks", its a difference largely orthogonal to the issue of protected class status under fair housing laws.
That's a straightforward First Amendment issue.
"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances."
You can't prohibit neo-Nazis from peaceably assembling. You can't prohibit people wearing purple shirts from peaceably assembling. In neither case does that make neo-Nazis nor purple-shirt-wearers a protected class.
The comparison between Nazism and discrimination against black people is pretty creepy, though.
The experience of black people on Airbnb is: "Here is an amazing looking listing. I'm psyched to see if it's available for my trip. Click. Oh! It is available! How excellent. I'll plan accordingly." Then, a few hours later, when the owner has noticed the applicant's African American sounding name, the reservation is rescinded. It happens repeatedly, and there's nothing the renter can do about it.
The experience of the neo-Nazi is roughly the same, except the first time it happens they take the swastika out of their profile picture and from then on everything works peachy for them.