It is an inaccurate word, and an insulting one. There may be a person or two out there who has some sort of “phobia” about people who have changed from their biological sex to one of the other genders – they may feel they are going to have seven years bad luck if they see one, or believe that such people are out to get them – but I have yet to meet any. Like the word “homophobic” before it, it is meant to suggest that there is something psychologically wrong with the person who disagrees with you. Those disagreeing must be secretly afraid that they are homosexual, or have some other sexual wrongness, unspecified.
That’s pretty insulting. It is an effort to avoid logical discussion and poison the well against another person’s opinion. These "phobias" not something measurable that can be tested. It is a word with no precision, merely suggestive, as it is used against anyone who the speaker feels doesn’t have quite the right beliefs about people who have transitioned, or wish to. It is rather like saying that people who were “against the war” are liberals. It is a generalization that doesn’t hold. Some conservatives and many libertarians were and are against wars in the Middle East. Some liberals, especially elected officials, were for it.
Or also, it is like calling all Muslims “jihadists,” or people who wish to reduce illegal immigration “racists.” Even if some of each are, it is a smear attempt to attach that to others.
It is ironic that a movement that is becoming increasingly strident that pronoun choices can be insulting would choose to be insulting in a similar manner themselves. Or perhaps not ironic. They believe in the power of language to shape a discussion, and attempt to preempt a discussion by setting the initial terms in their favor.
Words do gradually change in meaning. Phobic and –phobe have recently taken on the meaning of “dislikes” in addition to “fears.” This has been calculated, because the emotional tone of the derivation remains in play. Not all dictionaries even note such a meaning, but others do. However, even those that do first regard it as metaphorical, a poetic exaggeration, as in “I’m phobic about buying anything that’s advertised with hearts on it,” meant as a wry joke against oneself. The popular usage of “-phobe” as “one who dislikes,” is very new, and needn’t be regarded as a shared bit of language, just because others would like it to be. If the term persists, then in fifty years or so one might fairly say that the origin is no longer part of the meaning. Happens all the time in language. But it hasn’t happened yet, and we still depend on words to convey denotative as well as connotative meaning.
It is okay to challenge these terms on the basis of inaccuracy and insult and ask others to refrain from using them. Not that we need to do this at every turn, lying in wait for any unwary soul who trespasses against us. In many cases, people run in circles where that is simply the word used, and they don’t think about it much. We all like language shortcuts and few of us are precise. Jumping all over a person for using the word “transphobic” might alienate a person who largely agrees with you, once they think about it. To those who would insist “But that’s what they are. They deny the identity of trans people. That is hateful,” I think it is fair to point out that a) it is a generalization, and b) it is insulting. When asked* what word should be used instead I would suggest that people not use a single word, but a phrase or even a sentence, describing exactly what they mean. If that seems cumbersome, because people aren’t used to it, well, adjusting to new pronouns is cumbersome as well.
*I am being kind here. It won’t be “asked.”