Yes, the young character known affectionately as "Egg" while traveling incognito as a humble knight's squire is indeed Prince Aegon Targaryen, who eventually ascends to the Iron Throne as King Aegon V Targaryen. Because he was the fourth son of a fourth son, his chances of wearing the crown were initially so minuscule that he earned the historical moniker "Aegon the Unlikely" before unexpected successions and Great Councils ultimately positioned him to rule the Seven Kingdoms.