In the specific contexts in which the character is recognised - first at a tournament and second in the streets of London - he would have been pretty noticeable, but in a wider context the answer is 'no'.
At a tournament, he would be in the company of his own nobility. In the film, this is what Edward was worried about. He didn't want people to go easy on him or forfeit on account of his royal blood, and wanted royalty to be treated like any other competitor. In reality, Edward's father, Edward III, actually did this by having his tournament team all wear identical kit with helmets that covered their faces to reinforce the idea that royalty aren't anything special at a tournament. You can read more about that in an answer here by u/serMontagu. It would have been hard for his competitors not to know his face, given that they'd fought alongside him in the Hundred Years War with Edward routinely leading contingents of armies, and sometimes entire armies. They would have interacted a lot; in battle, at banquets and feasts and other courtly functions, and of course at other tournaments. The odds of other noblemen recognising him by just his face would have been very high.
London always had a special relationship with the crown. Kings were coronated there, the city was officially the possession of the king (though in practise it was self-governing), and although the relationship between the city council and the king had at times been very difficult (Richard II had revoked the city charter and they threw a massive party to try and get him to give it back) it was pretty good during The Black Prince's lifetime. The Black Prince enjoyed London; he visited the city frequently, held parades there when he came home from battle victorious, and it's where he eventually died. His father enjoyed it more, which meant he was a common sight at special events.
In 1359, Edward III and Edward the Black Prince hosted a tournament in London. This was a three day affair to mark a royal wedding, and seems to have been done to emphasise the strong relationship between the city and the royal family. So when the mayor and the sheriff opened the tournament, they got up in front of the crowd and the crowd went wild, recognising the pair for who they really were. They were not the mayor and sheriff, but Edward III and Edward the Black Prince. The crowd was expecting them, of course - this was to mark a royal wedding and it would be a bit odd if the king wasn't there - but that he was participating and pretending to be the mayor was apparently a surprise. Then the noblemen fought in costume as various city officials before the jubilant crowd. They probably recognised Edward III more than they did The Black Prince, because Edward III spent more time in the city, and it was far from the first London tournament he had hosted.
But more generally speaking, probably not. Being recognised was a problem pretty specific to Edward III and his son, and even then only at tournaments or in the one city they happened to spend a lot of time. Medieval kings couldn't put their image out there in the form of any mass media - the closest they had to that was a sermon and it's quite hard to describe the exact details of someone's face good enough to be recognisable. They could recognise that they were noblemen of some kind from their clothes and equipment and their entourage (if they were travelling with one), but unless they were announced then they were somewhat unlikely to be recognised specifically as royalty. If the nobleman was local a peasant might be able to recognise their coat of arms too, which was the only recognisable symbol individual nobles could display publicly. By this point the royal family was adopting this as a symbol, so if they were waving that about they'd be seen, but then that was the point of it.
So to be able to look at a crowd of people and go 'hey, there's the king!' was not a level of familiarity your average peasant would have had. It just so happens that the two times he goes in disguise in the film are perhaps the only circumstances in which the real Edward might have been recognised by appearance.
Sources:
Barber, Richard, and Juliet RV Barker. Tournaments: Jousts, Chivalry and Pageants in the Middle Ages. Boydell Press, 1989.
Barron, Caroline, Martha Carlin, and Joel T. Rosenthal, eds. Medieval London: Collected Papers of Caroline M. Barron. ISD LLC, 2017.
Crouch, David. Tournament. A&C Black, 2006.
