He could discover it through some kind of quirk inheritance test. In the fanfiction world, there might be a scientific or magical test that reveals his true quirk inheritance. So when he takes this test, it shows that he has his father's quirk.
In some fanfictions, Izuku might discover he has his father's quirk during a moment of extreme stress. For example, when facing a powerful villain that he couldn't defeat with his own quirk, suddenly a new power, which turns out to be his father's quirk, emerges.
It might also change his relationships with other characters. His friends could be both amazed and a bit envious. And his enemies would start to see him as an even greater threat. For example, Bakugo might feel more pressured to compete with Izuku who now has an additional powerful quirk.
In 'The Healer of UA', Izuku's healing quirk is more complex. It requires him to concentrate and visualize the injury being healed. He can also transfer some of his own stamina to the injured to speed up the process. This makes his quirk not only about physical healing but also about his own sacrifice and determination.
Some characters might be immediately terrified. They could avoid Izuku or even try to drive him away. For example, classmates who don't understand his quirk might spread rumors about him.
In fanfiction, Izuku often has unique ways to cultivate his quirk. One common scenario is that he focuses on intense physical training. Since his body was initially not as strong as it could be to handle his powerful quirk, he might do a lot of strength - building exercises like weightlifting and running. This not only helps him control his quirk better but also increases his overall combat capabilities.
His relationship with his mother could become more complex. She might be worried about his new nature and try to protect him more. With his friends, like Bakugo, they might initially be scared or distrustful when they find out about his quirk.
In many fanfictions, Izuku might unlock his quirk through intense emotional experiences. For example, a moment of extreme anger or determination could trigger it. Maybe seeing his friends in great danger and feeling a strong need to protect them could be the catalyst.
In some fanfiction, Izuku also experiments with different environments to cultivate his quirk. If his quirk is related to strength, he might try training in places with different gravitational forces. This forces his body and quirk to adapt, and in the process, he can discover new ways to use and develop his quirk.
He could have the power of flight. Just like a dragon, being able to soar through the sky would be an amazing ability. He might also have enhanced strength, as dragons are often depicted as very strong creatures in mythology.
The Gamer Quirk also changes Izuku's relationship with others. He can now offer unique insights based on his quirk. For instance, he might be able to analyze a villain's attack pattern like it was a boss fight in a game. His friends and allies can then use this information to their advantage. This makes him a valuable member of any team. At the same time, it also means that he might be seen as a bit of an oddball because his thinking is so different from the norm. But overall, it helps him form deeper connections with those who understand and appreciate his quirk.
His relationship with his teachers could also change. The teachers might be more interested in studying his sentient quirk, which could lead to special training or extra attention. This could either make Izuku feel special or pressured, depending on how the author writes it.