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The difference is, "dog-hater" is not a protected class as defined by public accommodation laws, so they would not have any grounds for a lawsuit. Sexual orientation is, at least in some states.

Legally, the baker would not be "wrong" in your scenario, though in my opinion they would at least be foolish. Defiling a cake does not harm a dog and once the customer pays for, it is their property to do whatever they want with.

How about if your dog-hater were driving late at night, they got tired and pulled over into a motel. They're signing in for a room and the topic of dogs comes up for whatever reason. They mention that they hate dogs and the motel owner says "Get out, we don't allow no dog-haters in here". Are they right to do that? Legally, again, they could, though they couldn't on the basis of sexual orientation (in some states) or race, religion, national origin (everywhere).

The Supreme Court has already sided with the baker on the basis of the cake being an artistic expression, so it's a moot point now.