That's not what this is about. In both cases, the state is forcing an company to do something they don't want. The only difference is that you agree with the one but not the other.

Yes, because the difference is that they are essentially immutable personal traits. The parent was presenting this scenario as an equivalent one and I don't think they are.

How is religion any more or less immutable from political beliefs? In my view both are fully equivalent.

Religion is a protected class to stop zealots from waging wars on each other, got to nip it in the bud before tension builds up. It is a very rational reason even if the protection shouldn't be needed in theory.

In some cases (e.g. Jews, Amish) the lines between race/religion/national origin become pretty blurry. I agree that it's the most ambiguous of the classes, though.

Personal traits are always mutable. What seems (and may be) immutable to you could be mutable to others.

It's technically true that you could find some way to dye your skin a different colour, but I don't think most people would find that to be a realistic solution to anything.

It is in some cases (such as employment) in certain states. Ironically they were often enacted by left wing legislators, but are now being used to defend right wingers.