The editors of the Associated Press Stylebook discourage the use of the word homophobia, and this is their explanation: "Phobia means irrational, uncontrollable fear, often a form of
mental illness. In terms like homophobia, it's often speculation. The reasons for anti-gay feelings or actions may not be apparent. Specifics are better than vague characterizations of a person's general feelings about something."
Their ruling on this point in reasoned, principled, and wrong-headed.
An article at Slate quotes George Weinberg, the psychologist who popularized the word in 1972, as why he used the term phobia: He identified people displaying an strongly vehement dislike of gay people. "They had no argument, just repugnance. They felt this way even about their own children. I realized this thing is deeply emotional and is based on fear.” This is a dislike and discomfort of uncommon intensity.
The stylebook editors are quite right to say that it is not a word to be used casually, though it sometimes has been. In the same way, racism and misogyny got thrown around a good bit in the mercifully concluded presidential campaign. But to say that you should not casually call someone homophobic or racist or misogynistic is not quite the same thing as saying that you can't talk about homophobia or racism or misogyny, especially when ample evidence is provided.
Moreover, as the estimable Ben Zimmer points out, it may be overstating the case to identify phobias so narrowly with mental illness: “Words ending in ‘phobia’ are commonly used outside of clinical contexts. You can think about the word ‘xenophobia,’ which has been around for more than a century to refer to hatred of foreigners. That’s not a clinical condition in the same way that ‘homophobia’ isn’t a clinical diagnosis.”
You could, for example, go to The Phobia List and decide how serious a mental illness you want to call atelophobia, the fear of imperfection, or xanthopobia, the fear of the color yellow.
I myself side the the clear-sighted Doug Fisher at the University of South Carolina, who remarks of the AP policy: "While I like the concept, good luck with that. '-phobia' has become one of those utilitarian phrases people like to glom onto things to create neologisms. I think this is like trying to stop sand in the desert, and I doubt it will be long before we see it on AP's wires, having gotten past some eagle-eyed editor who has heard the phrases in widespread use."
And there is the heart of the matter. The word homophobia has been in widespread use over the past forty years. In some contexts it identifies a revulsion so irrational that it amounts to a mental disorder; in others, it identifies a pronounced dislike with cultural and political elements. It gets used because it is useful in describing an identifiable phenomenon.