Well, Snape's care can make Draco more confident. Knowing he has someone like Snape looking out for him gives him a sense of security. This can lead to Draco being less arrogant and more self - aware. He might start to question some of the pure - blood supremacist ideas he was raised with, as Snape's care could show him that there are more important things than just blood status.
There could be a situation where Harry is being bullied by some students from other houses. Snape, despite his usual cold exterior, steps in and scolds the bullies. For Draco, if he gets into trouble with other Slytherins over some internal house politics, Snape would quietly resolve the issue. In addition, Snape might offer to tutor them personally. He would give Harry extra help with his Patronus charm, knowing its importance, and for Draco, he could teach him some advanced potions techniques that are not taught in regular classes.
One common scenario is Snape protecting Draco from the other Death Eaters. Since Draco is young and somewhat naive in the ways of the Dark Arts and the Death Eater world, Snape steps in to keep him safe. Another scenario could be Snape helping Draco with his schoolwork, especially in Potions. Snape, being a master of Potions, would be able to guide Draco and boost his confidence in that subject.
Harry is known for his kindness and loyalty. So, he would take care of Draco with a lot of patience. He would probably stay up all night if Draco was fussy, just to make sure he was okay. His protective nature would also mean he'd keep Draco safe from any harm, whether it's from other wizards or just from the general dangers in the wizarding world.
One common theme is redemption. Draco often has a chance to be redeemed through Harry's care. For example, in many fics, Draco starts to question his previous actions and beliefs.
Snape is a very complex character. He might use his knowledge of magic and the Hogwarts environment to his advantage. For Harry, he might keep a closer eye on him during classes, subtly guiding him without showing too much favoritism. With Draco, being from Slytherin, Snape could offer more direct advice and protection. He would have to be very careful in public to not seem overly partial. He might also use his position to give them both extra tasks or privileges that help in their development while still maintaining the appearance of fairness among all students.
It's popular because it creates an interesting dynamic. Snape has always had a complex relationship with Harry in the original series. When Harry is deaged, it gives Snape a chance to start anew and perhaps be a different kind of influence. It also allows for exploring Snape's more nurturing side which is a contrast to his usual cold persona.
One possible plot could be that Harry discovers a time - traveling accident has brought baby Snape to his present. Harry, being the kind - hearted person he is, takes it upon himself to care for the baby. He has to figure out how to feed the baby, keep him clean and comfortable. Maybe he also has to protect baby Snape from any threats in the wizarding world that might target Snape even as a baby.
One popular 'harry takes care of draco fanfiction' is 'The Green-eyed Dragon'. In this fic, Harry discovers Draco in a difficult situation and decides to look after him. It's really heart - warming as it shows a different side of their relationship, with Harry being protective and Draco gradually opening up.
One possibility could be that Draco is forced into taking care of Harry due to some magical circumstances. Maybe there's a spell gone wrong that makes it his responsibility. He initially resents it but as he spends time with baby Harry, he starts to see the innocence in him and develops a soft spot.