I think if you put a large number of people in this experience, not all of them would feel humiliated. Therefore, it's inappropriate for a journalist to declare it humiliation as a capital-T Truth. What's the most appropriate word... humiliation, offense, insult, annoyance, _____? I don't know. It's debatable and relies on individual and personal perspective, so it deserves to be in quotes.
For me, based on the woman's description of the events, I think humiliation sounds like a strong reaction. I have to base it solely on what she wrote, because I wasn't there. The article states they were "repeatedly" asked, but the original post only identifies that they were asked twice. The only "demand" was to see a boarding pass (a pretty routine and, I think, legitimate request when trying to settle seating arrangement requests/disputes). We don't know if the couple had the same last name, so it wouldn't be immediately obvious on paper they are a family unit. The post says "three consecutive seats," but they were split across an isle and many people could think that isn't truly consecutive seating. The "Flight Manager" probably wasn't the same person as the "Customer Service" staff from earlier in the post (or we wouldn't call them two different names), and we don't know what kind of information they were operating off of. As readers, we're missing a lot of information and context in the storytelling (which is entirely one-sided). I'm not ready to light my torch and turn the safety off on my pitchfork.
From the point of view of the airline... they have a set of passengers who have made a request, and they are trying to find a way to honor that. They asked another passenger to move. Perhaps they made some false assumptions about who they were asking, but in the end, no one was forced to move. I've been asked to move when I was sitting next to a friend. Why should a married couple have greater rights to assigned seating? They don't... but they made a request and the airline tried to accommodate them. As a passenger, it is my right to politely decline that request. As a business, it is their right to try to convince me to change my mind. I can stand my ground or acquiesce. That's my choice.
Now, if someone said, "Hey, you, you disgusting lesbians... you aren't as important as a real married couple! MOVE ASIDE!!!" that would be a completely different issue; I imagine most people would agree that is a humiliating experience. This is hyperbole, of course, to prove a point. In the end, as the story was described, I doubt that would qualify as a humiliating experience in my life, and presumably the same for many others, so the quotation marks are justified.