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Iomedaean Sword Oath, is it a feat? It's worded like a feat, has the same format, but is it a feat?
Yep, it is a feat. It lets your cleric (or whatever non-fighter class you're playing) specialize in the longsword. In theory you could gain access to other feats someday, but so far the only official "longsword-related" feat published that requires you to have four fighter levels is Weapon Specialization.
You're spending a feat to be able to take a different feat that your class normally cannot take, and that +2 to damage is pretty nice. I'm not saying it's the greatest feat ever, but I can see why you might want to take it if you were a more combat oriented cleric. Or a devout rogue. Or an inquisitor or something.
Like you say, it looks like a feat. And the "Special" section certainly is a good indication that it is a feat:
Special: If you ever use a melee or ranged weapon other
than a longsword in combat, you lose the benefits of this
feat until you receive an atonement spell. You may still use
spells that act as weapons (such as flame blade and spiritual
weapon) without affecting your oath.
For further clarification look here. ;)