JAMES: I only said he might be in Slytherin. (1.1.4)
It would oversimplify the sorting process if you thought that your house defined your personality instead of the other way around. Albus is too young to realize this—as Harry once was—and he thinks that being Slytherin might make him evil. It's a self-fulfilling prophecy of sorts.
HARRY: Albus Severus, you were named after two headmasters of Hogwarts. One of them was a Slytherin and he was probably the bravest man I ever knew. (1.2.37)
Harry set high expectations for his son by giving him these two names. Even before Albus is given the label Slytherin, he's already marked—as Harry was with his scar—with names that give him a legacy to live up to.
HARRY: If it matters to you, you, the Sorting Hat will take your feelings into account. (1.2.39)
Is it possible that the Sorting Hat knew that Albus not only belonged in Slytherin, but wanted to be there as a subconscious way of subverting his father's expectations? It's a play, not a novel, so we can't hear Albus' thoughts. We wouldn't be shocked to learn if he thought, I'll really stick it to dad if I get sorted into Slytherin?
SCORPIUS: But, on the whole, I'd rather be a Malfoy than, you know, the son of the Dark Lord. (1.3.45)
You have to do what you can to come to terms with your identity. You can try to change it, or you can accept it. Scorpius accepts his name because it's better than the rumors that he's Voldemort's son. Perspective is important.
ALBUS: So what would you like me to do? Magic myself popular? Conjure myself into a new House? Transfigure myself into a better student? Just cast a spell, Dad, and change me into what you want me to be, okay? It'll work better for both of us. (1.4.117)
If it were possible for Harry to do these things that Albus sarcastically suggests, do you think he would do it?
RON: You persuaded the Sorting Hat, don't you remember? Panju bet you that you couldn't get into Gryffindor if your life depended on it, so you chose Gryffindor to spite him. (2.9.25)
Here, Ron explains how Albus ended up in Gryffindor in an alternate timeline. Choosing a House to spite another student sounds like something a Slytherin would do, not a Gryffindor, which makes us think our theory about Albus choosing Slytherin even more true.
DOLORES UMBRIDGE: No need to be modest, Scorpius. I've seen you on the Quidditch pitch, there's rarely a Snitch you don't catch. You are a highly valued student. Valued by the faculty. Valued especially by me. (3.1.6)
In the "Harry Is Dead" world, Scorpius is pretty much Harry Potter, but better at homework (because he pays someone to do it). Scorpius is the son that Harry wishes Albus was, and he's living the life Draco always wanted him to have. But that's the life that Draco would have wanted for himself. He doesn't push Scorpius the way Harry pushes his own identity on Albus.
SCORPIUS: The "Mudblood" death camps, the torture, the burning alive of those that oppose him. How much of that is you? Mum always told me that you were a better man than I could see, but this is what you really are, isn't it? A murderer, a torturer, a— (3.3.17)
Which Malfoy do you think is the "real" Malfoy? Is he the concerned dad in the original timeline, or the power hungry maniac in the bizarro timeline? Could he be a little bit of Column A and a little bit of Column B? Will the real Draco Malfoy please stand up?
SNAPE: Strange, isn't it? What comes from within. (3.9.81)
Snape and Scorpius know that your identity isn't necessarily what others define you as. They defy expectations. Harry, however, conformed to expectations, and he expects his own son to do the same.
SCORPIUS: I am—Scorpius the Dreadless. I am—Malfoy the Unanxious. (3.14.7)
Albus often wishes he wasn't Harry Potter's son, which means he would have a different name. Scorpius, however, embraces his name and elaborates on it. Without his father's upbringing, Scorpius wouldn't be the young man he becomes at the end of the play, and he acknowledges this.