In some fanfictions, it could be a result of some magical influence. Maybe there was a spell or a charm that made Harry act this way. Or it could be an expression of deep - seated emotions that Harry has been suppressing for a long time. This action then becomes a way for him to break free from his self - imposed restraints and reach out to Draco in a very physical way.
Perhaps Draco has developed a sense of respect for Harry over time. He might have seen Harry's courage and determination up close and decided that Harry is worthy of his help. Throwing the wand could be his way of showing that he acknowledges Harry's importance in the fight against the common enemy.
Well, it could be that Harry, at that moment, saw Draco not as his rival but as a fellow student. In the wizarding world, Quidditch can be a very dangerous sport. Harry might have thought about how his friends would feel if he were in Draco's position. Also, Harry has a heroic streak. He might have instinctively jumped into action without really thinking about who it was he was saving. His first impulse could have been just to stop a potential disaster from happening on the pitch.
Perhaps Draco realizes that Harry is in a dire situation where he needs the wand to save the day. Draco, having a change of heart, decides to give Harry his wand as a way of contributing to the greater good. This could be related to a looming threat that only Harry can face, and Draco understands that his wand can be a crucial tool for Harry.
One possible reason could be their long - standing rivalry. Draco has been taught to dislike Harry from a young age due to their family differences. His pure - blood supremacist upbringing might make him reject Harry as a mate because Harry is seen as a 'Mudblood - lover' in the eyes of Draco's family and their circle. Another reason could be fear. Draco might be afraid of what the wizarding society would think if he accepted Harry as his mate. The wizarding world has its own set of norms and expectations, and a relationship between them would be highly unconventional and likely face a lot of opposition.
Perhaps in a parallel universe where the whole wizarding world is different. There are no enmities between the Slytherins and Gryffindors from the start. Draco and Harry are friends, and Draco sitting on Harry's lap could be a common occurrence during their friendly banter or while sharing secrets in a cozy corner of the Hogwarts library. It could be a sign of their deep - seated friendship and trust.
A possible theme could be forgiveness. Harry might be in a position where he has to forgive Snape for all his past misdeeds. And this rather strange act of Snape bouncing him on his lap could be a catalyst for that forgiveness. Additionally, it could also touch on the theme of growth and change, both for Snape, who is showing a different side of himself, and for Harry, who has to adapt to this new side of Snape.
One reason could be for a disguise. Maybe he's on a secret mission and needs to pass as someone else. Another could be peer pressure. His friends might have dared him to do it.
It's also possible that Dumbledore has been misinformed or is under the influence of a dark spell himself. In the wizarding world, there are many forces at play. He could have received false information that made him think this was the best course of action for Harry. Another aspect could be that he is trying to teach Harry a lesson in a very extreme way. Maybe Harry has been too reckless, and Dumbledore wants to make him understand the seriousness of the situation.
One possible reason could be punishment. Snape has always had a rather thorny relationship with Harry in the original series. He might see this as a way to assert his authority over Harry. Maybe Harry has done something in the context of the fanfiction, like accidentally messing up Snape's potion ingredients or getting into an area of Snape's house where he wasn't supposed to be.
Maybe Draco says some really mean and hurtful words to Harry. For example, he might make fun of Harry's family situation, like his parents' death or his relationship with the Dursleys. Words like that can be really cutting and could easily make Harry cry.