Well this is Pottermore and yeah, it is possible for people to be Sorted into a house that they don't want to be in. Even in the books, the Sorting Hat takes your desires into consideration to an extent. If a student really didn't want to be a Gryffindor but would do well in Hufflepuff, the Sorting Hat would simply pop them into Hufflepuff unless it had a really strong reason not to.
Neville wanted to be in Hufflepuff, but the Sorting Hat disagreed. It saw something in Neville that 11 year old Neville did not yet see in himself. So Neville got Gryffindor anyway. But for the majority of more ordinary students who weren't destined to be heroes or fight in battles, maybe the Sorting Hat would be a little more lenient.
Everybody has a bit of each house in them. The Sorting Hat just tries to pick which values are most important to you, rather than what you actually are. So even if you were not the brightest crayon in the box but you still valued learning and intelligence above all else, you could wind up in Ravenclaw. If you're the most cunning, resourceful and Machiavellian fucker there is but you still find fairness and equality more important, you'll go to Hufflepuff rather than Slytherin. At least that's how I view it and I like my system better. People should be in the houses for the traits they value and strive towards.
It's not possible to re-Sort people in the HP universe, but in real life? That's a different story. The Sorting process is so fucked up anyway, pegging the entire human population into 4 groups. Why shouldn't a re-Sort be possible? It shouldn't be: "Oh, Pottermore put you into Slytherin, so that's it. You're a Slytherin until the day you die!" That would be ridiculous. People change all the time.
In Hogwarts it would be unreasonable to re-Sort students, because in that school the houses serve not just as tenents of the traits they represent, but as a sort of student "clan" where students feel at home, make friends and work together to win the house cup. The houses would mean a lot less in the schools if you could just switch houses. It's like how many boarding schools in the UK work in real life. Once you're in a house, that's it. You're extremely unlikely to be transferred except for very special circumstances. In real life it's a bit easier though, since you're not allocated by a magical Sorting Hat but by a boring old administrator.
I dunno where I'm going with this. But yes, in real life nobody is just 1 house. We all have every house trait, to some degree or another. All of the traits (intelligence, bravery, cunning, loyalty, justice) are in every single human being, to varying degrees. So we are all every house. In that sense I think it's okay if you want to pick 1 house you wish to identify with more. I just roll with Slytherin because I like Slytherin, but like almost all human beings on the planet I'd be a mix of everything.