These companies are not connected by ownership, so it's just a total "coincidence" that they are retro script words. They are similar though — all three are photo/video sharing sites, each with a twist of their own. To be fair, the above Instagram logo is not their latest. They have since redesigned their logo. It's Soooo different now, see:
Of these three, I'd say the Instagram one has the least longevity. Somehow they went backward into future of the 90's with the redesign. It's not as clean and mod as the first and it's not far back enough to be cool. At least the other two have customized to indicate their product — Pinterest with the pinpoint on the P and Vine being one connected vine.
I don't know exactly what it is that causes creative industries seem to come up with multiples of one theme at the same time, but I wouldn't call it coincidence.