I understand bigotry to be intolerance to contrary opinions, views, etc.
I wouldn’t assume the person is either. They may not be intolerant to you holding your beliefs, or whatever, but take offense at your offering of them.
An equivalent might be an atheist who takes offense at the expression ‘god bless you’ when you sneeze. They aren’t necessarily opposed to you believing in your god, but your conviction places them in an uncomfortable situation.
They don’t like the casual statements that imply that your god objectively is true and all that you believe about it is true, because it comes across as arrogant. You treat your chosen truth as if it is objectively discovered to some degree, and others are supposed to respond with kind sentiment, but at the same time by doing so, they further support that idea they hold to be in error. In fact, many simply react to being put in the predicament in the first place, and blame the merry wisher for doing so (i.e. they treat their chosen reality as if it discovered fact, and they expect you to act in kind).
This person perceives a world of sentiments based on false precepts, and is usually offended by any notions expressed to its support.
I think back to your question now and wonder where you draw the line between bigotry and intolerance. Is taking offense to every utterance of contrary opinion (flbw) bigotry? If so, then they are likely a bigot by this definition. Is taking offense the same as intolerance?
Sorry to dive into the word game; it’s not for the purpose of arguing semantics. It’s just considering that however you define your two labels obviously dictates which would or would not apply.
To the spirit of your question (at least that which seems to me): You seem to be unsettled yourself at their taking offense. If not, then I assume you are in fact asking about the two terms themselves and if their respective definitions apply.
I personally treat all merry wishes the same, because I understand the person’s intention is to wish me well in a sincere and earnest fashion. I appreciate that regardless of what verbal form it takes. I even appreciate that I meant enough to them to even acknowledge me.
I think it’s an emotional response either way, and people can add whatever they want to their list of things that offend them. I try to keep my own list pretty short; after all, why would I want to compile a larger list of things to upset me.