From the text of the books, there's easily a lot more support for Varys being a Blackfyre supporter and that Young Griff is indeed (f)Aegon, i.e. a Blackfyre, albeit through the female line, i.e. descended from Shiera Seastar. Then again, Blackfyre was never really a "family name" in a traditional sense.
In fact, I think it is somewhat obvious that in TWOW, GRRM is setting up Aegon to conquer Westeros and "come out" as a Blackfyre. "The rightful ruler of the Seven Kingdoms, king of the Andals and the Rhoynar, first of his name (red flag... wait, shouldn't it be 6th of his name?) ... yada yada. I give you Ageon... BLACKFYRE." This will set up a second Dance, which would either occur towards the end of TWOW or the beginning of ADOS.
In the show, Varys looks to be currently supporting Dany's claim as a Targaryen, making him a Targ loyalist. This doesn't make too much sense to me, and would be a massive, somewhat lazy retcon in terms of allegiances and motivations, as portrayed in the earlier seasons. Unfortunately, this is starting to look likely.
My hope is that, in the show, Doran (and crew) are a proxy for Jon Con and Ageon Blackfyre is pretending to be Aegon VI Targaryen pretending to be Trystane Martell. This preserves most of the book's intent. Yes, there are some holes that would stem from this... for example, what the BLEEP was Illyrio Mopatis doing the whole time, then? But it's way cleaner than any other alternative. If this isn't the route D&D are planning to take, though, Yara save us all!