Much of my focus in this answer will be on the Black Prince's father King Edward III (1312-1377) who was alive for all of the Black Prince’s adult life. Additionally, while your answer focuses on the joust, much of the source material on tournaments in the reign of Edward III focuses on melees (mock battles between teams) as these were the most popular event in tournaments of the era. In short, based upon my knowledge of Edward III and tourneying, royalty absolutely were treated like any other (noble) competitor.

Tournaments, including jousting, were about more than individuals earning glory for themselves. They were most importantly an instrument for confirming and strengthening fraternal bonds between a lord and his servants. This rang true no matter the rank of the lord, whether it was a local landholder or the king himself. I would argue that no English Royal ever saw the value in this bond created by tourneying as Edward III who had tournaments organised for almost every major event in his reign such as his coronation, his wife's coronation, the birth of his children and to mark military victories. As such, Edward III would fight alongside his men against enemies who would treat the king as any other enemy in the melee. In fact Edward III loved the drama and pageantry of the tournament so much that he became infamous for the sheer amount he spent on bespoke outfits, disguises and masks for his entire team to wear. Sometimes this would take the form of identical uniforms such as the 1331 tournament at Cheapside in which his team all dressed as Tartars to promote the crusading movement and impress a visiting Armenian contingent. Other times the king actually wore a uniform in his team that placed him below the rank of his teammates with one outfit he liked to wear at tournaments that denoted him as ‘simple knight’.

In these team games it was also not a given that the royal would be the captain of the team. Edward III would often have two of his close friends WIlliam Montague (Earl of Salisbury) and Bartholomew Burghersh lead the team that Edward was on while he adopted his simple knight persona. This also meant that the whole team, including the king would adopt the arms of the captain of the team as well as the same uniform, making it incredibly difficult for an opposing knight to identify that they were about to strike royalty.

Tournaments and jousting were an incredibly important and valuable instrument in the reign of Edward III, and so the life of the Black Prince, for royalty to foster fraternal bonds between themselves and their men. Therefore royalty would be treated just like any other competitor and doing so was actively encouraged by kings such as Edward III to make those militaristic bonds even stronger.

Sources:

J. Vale, Edward III and Chivalry: Chivalric Society and Its Context, 1270-1350. Boydell Press (1982)

J. Bothwell, Edward III and the English Peerage, Boydell Press (2004) I.Mortimer, The Perfect King, London (2006)

W.M. Ormrod, England in the Fourteenth Century, Boydell Press (1986)

Charter Rolls, Fine Rolls and Patent Rolls on English Parliament.