My wife's cousin was a GIGANTIC Phish fan....even named his black lab Fee. He and his girlfriend would travel everywhere to see Phish. He was one of those guys that recorded all the shows he went to with a DAT machine and mini mics stuffed in his cap.
Never did he call them prog or proggy, he simply called them a jam band very much like the Dead....He said they were the Dead of the 2000's.
That's how I always saw Phish......Long jams always harken to prog rock long jams. Now I can stretch my thinking and agree they could have influence from the Softs, KC, Zappa and the like. If we are simply talking a label
then I would probably go with prog, that being said then so is the Grateful Dead.
My wife's cousin loved their music and that was enough for him......Me personally I would not label them prog in my collection......Fusion Rock or Rock Fusion, something like that in my collection.