@KateGregory has handled the title question quite well, so I'm just responding to the question in the body of your post.
You could probably string together a series of Greek or Latin root words and call it a new phobia if you want (I'm guessing this is how Coriat came up with "triskaidekaphobia", and that he was not alone in using that method)...but first of all, phobias are usually irrational, and given your history, your aversion to the middle seat does not sound irrational. (And you have my sympathy!) I always avoid middle seats too; I usually look for the most isolated seat I can find, preferably toward the back so everybody else's loud mouths are pointed away from me. Nothing irrational about this. People are annoying (and I somehow got my doctorate studying them anyway).
Second, if you think you really do have an intolerable reaction to this situation and can't keep it together if you find yourself there, it's possible you do have something personal to worry about. All else being equal, it's not fair for you to request special treatment, because chances are very few people wouldn't rather sit somewhere else too, and we can't all avoid the middle seats of course. If this doesn't address your reason for asking, maybe it would be worthwhile to talk to a psychiatrist. I'm not one myself – I'm just a psychologist specializing in ordinary personality variation – so I can't say that it's impossible for you to have a specific, legitimate phobia about this, or that it's impossible for you to develop some form of posttraumatic stress or social anxiety disorder just due to your legitimately awful experiences. It doesn't seem likely, but I can say that vulnerability to developing a disorder varies across individuals, so arguably no one can rule it out completely (see the diathesis-stress model for more on this principle).
Of course, there's little or no harm in seeing a therapist one way or another. They're not going to haul you off in a white straightjacket or force-feed you pills, and if they recommend any, you can always refuse (and you should always study any medication you're prescribed for yourself, IMO). Everyone can benefit from a little counseling, including normal, healthy people, which might very well include you (I'd like to say it probably does, but I don't want to go anywhere near a personal diagnosis in this answer). If you have insurance, the visit will probably be covered.
Aside from PTSD and social anxiety, there are other possibilities that would be nice to rule out (e.g., agoraphobia). This is the benefit of actually going to see a professional about it: you can get a clean bill of mental health if you really are completely healthy, and maybe even some useful tips if you're not quite there but definitely subclinical. Therapists are full of useful strategies for managing emotions that everyone could probably stand to know more about, disordered or not.
Forgive me for ending on a relatively grim note, but I wouldn't want to be accused of not taking your concern seriously. Until you see a professional, you also can't really rule out the possibility of a real disorder affecting you. This isn't to say, "Everyone and their mother should go get checked out right away, because you never know!" but if you've got some tangible, concerning signs, don't ignore them – you wouldn't ignore a physical symptom of something serious potentially affecting your heart, for instance, so why be any more cavalier about your brain? Again, I don't hear anything that sounds truly diagnostic or threatening here, but I'm not qualified to give you a clean bill either. Some sites on the internet will try, but whether they should is debatable (see "Finding help with psychological and emotional problems" on Meta CogSci), and I wouldn't want to go there even if I was qualified.
Best to see a professional in person if you feel there's a serious risk of panic attack or any other kind of unusual harm or loss of control afflicting you if you find yourself in the middle seat again. As @NateEldredge's comment indicates, you could seriously disrupt the operation of the flight if you let yourself get too out-of-sorts. Of course, if you do have special psychological conditions that make this situation exceptionally difficult for you, this is not an "all else being equal" kind of scenario, and you have every right to request special accommodation. Whether you'll be granted it is another question of course (or at least a part of yours that I can't answer)...but (this might be irresponsible of me to say) so is whether they'd actually do anything to verify it if you claimed to have a special vulnerability. I certainly hope they would take you seriously...but people with social anxiety disorders don't walk around with cards in their wallets for the sake of proving it to people. If you do get a positive diagnosis at any point, you might want to ask for special documentation just so you don't have to worry about being dismissed outright. If you don't get diagnosed (which sometimes happens even to people who should), at least you know some abnormal psych jargon you can throw at a flight attendant to get your point across.