Reread this.
The Harry Potter houses are a bad system for an IP-scrubbed roleplaying game. Consider that in Harry Potter itself, there are:
House Hero: Chosen Hero, Clever Hero, Chump Hero
House Villain: Clever Villain, Chump Villain
House Clever: Clever Extra
House Chump: Chump Extra
with Chosen, Clever, Chump being character classes (obviously unfit for a roleplaying game: Chump is Monk-tier, and Chosen plays Fate instead of D&D).
JK Rowling could get away with it because she was writing single-author fiction. Harry Potter roleplayers can use the House subdivision (but not the character classes) because they have huge swaths of material to serve as foundation and foil for whatever characters they want to play.
Having your players all be from the same Harry Potter House / Lo5R Clan / Planescape Faction is completely out of the question. Look at the OSSRs of the latter two to see why. People like different stuff. Furthermore, players gravitate to different character archetypes. If your Houses / Clans / whatever are described as even more important within the school environment than character classes, player will try to pick different ones. If your IP-scrubbed game advertises itself with WOW LOOK AT ALL THOSE THINGS YOU CAN BE!!1! AWESOMESAUCE!!1! and then immediately shut people down, they're going to be offended, and rightly so.
You can only demand that all the players be in one faction if it's a side in the main conflict of the game (because it's a cooperative roleplaying game and they all have to be in the same party trying to win). In a World of Warcraft Alliance vs Horde campaign, you can tell players to pick Alliance or Horde as a group (but you're going to have someone playing a Horde-aligned Night Elf or whatever anyway). In a campaign where you're killing demon cattle or demon calamari or Arabian bugs or whoever's Team Monster now, you'll probably get a better game out of allowing them to pick a faction individually. The only reason you can have for limiting players to one Harry Potter House is if the goal is to win the Cup, but the actual Harry Potter books make it really damn clear that the Cup is a bullshit goal, and your players will expect a more meaningful quest, too.
But more importantly, neither your Houses nor any but one of the Harry Potter houses can carry a character story.
"I'm from House Blaster but I'm not a Blaster-class" -- so what? What do Blasters do? Fight duels? Build mana reactors? If it's something that Blaster-classes are better at, then House Blaster second fiddles are not a real character option. If every House has its corresponding class, then you've just narrowed actual character choices to 6 options, which means that in the Game Design Flowsheet you can barely support a 6-player party -- and not well at that, because you already have House Chump, so #6 is going to defect, and, knowing this, #5 is going to defect. The two last picks are going to be Fred/George pairs to previously picked characters. Fred-or-maybe-George is
not a good player character, but 2/3 of your party are that. Congratz.
The only House that actually works okay is Slytherin, because they're also House Blood Supremacist, and in a brand-name Harry Potter game with Blood Supremacists being main villains, there's room for several Slytherins with different prompts (I'm an asshole but I'll get better / My parents are assholes / My parents
owe an asshole / I'm
secretly descended from an asshole and now everyone thinks I'm evil / My vampire boyfriend dumped me for Hilary Duff). All your Houses need to multiply character concepts, not subsume and diminish them.
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I give you that: following the legacy of D&D, you can actually skate somewhat on "we're friends and we're going to the same school". In both D&D and Shadowrun, the player character motivation to adventure together, to take jobs that come their way without too much scrutiny, and not to fucking betray each other is externally imposed. But after that's settled, they actually have jobs to proactively do. So you're friends, you're in the same grade, and try to not get turned into newts or, worse, expelled (you'd rather be newts than custserv reps).
Now you still need to decide
what the fuck it is that you're doing as a group in roleplayed scenes with all of the party present and contributing roughly equally, and then to ensure every character has the ability to meaningfully contribute (and then for the faction system to interact with it). Be specific, describe sample scenes. Write out an Example of Play if it helps. Recall that, e.g., for Shadowrun, "industrial espionage" is too vague and ends up allowing a hacker to play his own unrelated game without being present on location, which (surprise!) is bad for the game. If only one character plays Quidditch, you can't have Quidditch on camera. If it's somehow an indispensable part of being a wizard and everyone can play it, you can, but each character better have a role in the minigame to showcase their abilities rather than a "play Quidditch" skill tax. Whether you're preparing for the next big Test, or solving unconnected mysteries-of-the-semester, or hunting dark wizards, or overcoming random challenges (SURPRISE DISTRICT ASSESSMENT! PASS IT OR THE SCHOOL LOSES FUNDING!), you need to do it as a group.