"Happy Holidays" irritates Eugene. He says that the generic quality of the greeting gets under his skin; "Merry Christmas" is fine with him. But why should the generic quality of a greeting bother him? What could be more generic than "Hello"? But I bet that "Hello" does not bother Eugene.
I suspect that what annoys Eugene is that the person who says "Happy Holidays" to him is afraid of offending him. The person knows or suspects that Eugene is Jewish and worries that "Merry Christmas" would offend a Jew. Odd to think that you can irritate a person by trying not to offend him! But the key here is the presumption that Eugene is religious and, more to the point, thin-skinned about his religion. The person who says "Happy Holidays" to Eugene reveals that he believes that Eugene is thin-skinned and priggish and presumes that Eugene is religious when Eugene in fact is a relaxed and good-natured fellow and not religious at all (or so I assume). No wonder Eugene is annoyed.
The problem for the interlocutor who perhaps does not know Eugene well is that some people would be offended by "Merry Christmas." That greeting also presumes a great deal—namely, that the addressee is Christian or relaxed about religion or thinks of Christmas as a secular holiday. When that presumption turns out to be wrong, offense occurs. There is no doubt that "Happy Holidays" is a lower risk greeting than "Merry Christmas. The flight to generality occurs when people know each other less well and thus know less about their religious views and overall temperament. All in all, "Happy Holidays" is less presumptuous than "Merry Christmas;" so why does it annoy Eugene?
The answer is that more generic greetings (and other practices such as gift-giving) reflect social distance (as Eugene, says a way to "play it safe"); generic greetings should really only annoy when they are conveyed by people who are close to us. If a person who celebrates Christmas receives a greeting of "Happy Holidays" from his spouse or child or friends, the greeting would seem a bit chilly, and one would suspect that something is wrong. And if you celebrate and care about Christmas, a hearty "Merry Christmas" from a stranger sounds warmer than "Happy Holidays" because, it turns out, the stranger has something more in common with you than the stranger who says "Happy Holidays." The stranger is engaging in a high-risk, high-return strategy; a generic "Happy Holidays" is less likely to offend a non-Christian even if it sounds chilly.
Eugene might have had in mind someone who knew him well when complaining about "Happy Holidays," perhaps a colleague or student. Otherwise, he's being a bit hard on people. Or perhaps he is annoyed by the general tendency for people to adopt increasingly bland and, sometimes, euphemistic terms in order to avoid the risk of offense in all circumstances. The fear of giving offense—especially to ethnic and religious minorities, and people who do poorly for various reasons (uneducated, lower class, disabled, and so forth)—is ubiquitous in our society, but we should be used to it by now. Social distance is the price we pay for diversity. We avoid excluding some people by being remote to everyone.