Not according to Melisandre, who uses the two terms interchangeably (though she does use Azor Ahai more often).She's the Prince that Was Promised. Who is a separate entity from Azor Ahai.
The issue is murky because both Jon and Dany fulfill many of the prophetic requirements for being Azor Ahai/The Prince(ss?) that was Promised. Only Dany though has literally 'waken dragons from stones' if that line of the prophecy means 'somehow was involved in the birthing of dragons.'
That said, when Mel looks into the fire and asks R'hllor to show her a sign of Azor Ahai, all he gives her is Snow. And Dany's only had figurative rebirths, Jon has literally come back from the dead.
It's also possible they're both fulfillment of the prophecy, because it's quite often stated how prophecies are not accurate or too open to interpretation.