Spoiler Alert for supernatural activity and the Targaryen family tree beyond AGOT...
There seem to be a number of prophecies out there. Azor Ahai reborn. The three heads of the dragon. The Prince or Princess that was promised. The Stallion who mounts the world. The Horn of Joramun. The common denominator is that there will be a leader (or leaders) who will lead their people to fulfill their destiny to rule the world.. or or at least shake off the chains of oppression.
In addition to these prophecies, it seems there are a number of peoples who have this same expectation though they might not have a prophecy. The Ironborn expect a ruler to lead them to plunder the world. The Ghiscari look to a leader to return them to super power status. The Undying of Qarth desire a return to glory. And I don't know what the people of Asshai want...
A prophecy means that there is a supernatural power or divine being working to fulfill it. Without supernatural backing, how can a prophecy come to fruition? A prophecy implies a higher intelligence.
So what evidence do we have of supernatural power? The Wall. Warging. The powers of the priests of R'hllor. Others. Blood magic. Dragons. Bran and Bloodraven looking through trees. It's not overpowering, but it's there.
What about divine intelligence? First, let me just discount the existence of an uncreated creator in ASOIAF cosmology. Don't get me wrong, I'm not discounting an uncreated creator of our world... just that of ASOIAF. I mean GRRM could still reveal that the world of ASOIAF was created by a single omnipotent and holy being, but I doubt he will. Since there is no single creator, there must either be lesser divine powers or none at all. But we do see supernatural powers, so there must be some source to that.
Yet, I don't know that we've seen divine intelligence displayed on behalf of any people of ASOIAF. R'hllor's followers display power, but we only have their word for it's origins... and the actual displays seem more malign than benign. Necromancy, murder, human sacrifice...
So will the hidden divine beings show themselves?
The only prophecy seemingly tied to supernatural power is that of Azor Ahai reborn. Neither the three headed dragon, the stallion, nor TPTWP seem to have supernatural backing at this point.
So on to the original question...
Since when does Azor Ahai, and the whole "The dragon has three heads" thing.. Have to be the same thing? Where does it ever say in any of the books that Azor Ahai, and the Three headed dragon.. Have ANYTHING to do with each other? Just putting that out there..
TPTWP and the three headed dragon are both Targaryen prophecies. We know that Maester Aemon connected both of them with Dany before he died. That is the beginning of the linked prophecies. Now if the two go together, then we still need two more heads... and those heads must be of Targaryen blood.
Who are the other Targaryens out there? Aemon just died and Aegon just resurfaced. But from the example of Robert's bastards, there seem to be a few candidates who may be Targaryen bastards.
Jon is THE candidate. Jon ties in Ice to ASOIAF. He may be Fire as well, if Rhaegar is his father.
Other possible Targaryen bastards include Tyrion, Cersei, Jaime, Myrcella, Tommen, Darkstar, Aurane Waters, Brown Ben Plumm... But Jon is the real possibility... and Tyrion has an outside chance... the rest are just wild guesses.
What are the qualifications to become Azor Ahai reborn? To possess a magic sword of fire and light... and to have tempered his/her soul and sword by slaying his/her one true love. Well, Dany killed Drogo and birthed her dragons in his funeral pyre. And Jon was instrumental in killing Ygritte.
So Dany seems to fit Three Heads, TPTWP, and AAR. Jon may also fit all three. That's how I connect the prophecies... that's how some readers seem to think that they may in fact be one prophecy seen from different perspectives. If so... mayhaps there will be a greater divine being revealed.