He isn't a Blackfyre. Blackfyre was a specific Targaryen bastard who was legitimized and given the family sword. His legitimate brother inherited but was seen as weak and allowed Dornish influence into his court. Blackfyre though of himself as the only deserving heir of his father and eventually started a rebellion with one of his other bastard brothers. They lost, their sons ran to the Free Cities, plotted a return to Westeros, started a mercenary army, etc. but eventually all of the recognized heirs to the Blackfyre bloodline died.
Which is to say: Jon isn't a Blackfyre in any way shape or form.
He is a proper Targaryen, the legitimate or legitimized son of Prince Rhaegar and Lyanna. (probably)