Very academically able: eg for spgs, almost all pupils are in the top 10% and half are in the top 3% intellectual ability. It's very hard to get in: money is absolutely not enough.
Extremely hard working. The pp above who is at spgs probably doesn't even realise how hard they all work compared to most schools, because it's so ingrained in the culture and very much the norm. The casual mention of tutors - despite being in a school working at the speed that a class with only top-10% ability, well-behaved students can achieve - gives a clue.
Reasonably well-behaved and respectful of teachers, although many are quite sassy (but rarely very disruptive)
Very self-confudent. Since childhood, they've been trained by teachers - and usually also parents - to think critically, and their opinions are listened to. They've been given opportunities to follow their interests, and sipported in their efforts. As a result, they deep-down believe that they have a worthwhile contribution to make: that's the famous private-school confidence.
Very high expectations of themselves, which drives the hard work and also drives anxiety. This comes mainly from seeing so many successful, incredible peers. The lowest achievers in the class would be top-table in another school (since they're all top 10% academically) and there are a significant number of truly exceptional kids. So their personal bar for their own achievement is set incredibly high.
This drive is not only academic, but usually also in sports/music etc. Almost all will play at least one instrument and will have progressed through the grades: some will be grade 8 in several instruments. Almost all will be competent in sports and reasonably fit: some will play one or more sports at a high level, with external training camps as well as being taught at school.
They usually have intelligent, driven, successful parents with very high expectations of them. No, of course not all such parents send their kids private 🙄 - but it's pretty much a prerequisite to being able to (and choosing to) pay private fees. Only driven parents would choose an elite school like spgs for their DC: it's an incredibly hard-working, full-on environment for kids and only driven parents think that's worth the personal effort and investment from their children, to build the education, skills and strengths for their future.
They're also fun, interesting, normal kids! They'll have their own interests, love their families but get frustrated at siblings, help out at home willingly enough but would rather sit on the sofa,, adore their pets, are interested in the usual trends, worry about friendships - just like any other kids.