Do British schools make British people?

I've been going through one of those expat phases and thinking a lot about what makes British people British - and, in particular, what makes me feel British. A lot of these thoughts are coming back to schooling, which I guess is no surprise given it takes up the most part of one's formative years. I've narrowed it down to a handful of uniquely British features about British schools that I survived and now lead me to identify with other Brits as 'being British' -- whatever that means:

1. The words School Assembly make your arse ache. You spent multiple mornings aged 4 to 16 often sat cross-legged on a cold gym floor, singing hymns badly and occassionally offering up a polite but not overtly religious prayer for the school holidays to be extended or for the hall floor to warm up. In later years, you got to sit in chairs apart from the times when you were called up to perform some embarassing skit in front of your peers or read out the rugby/netball/hockey scores while the bad kids at the back jeered.

2. A 'playground' conjures up images of concrete swathes, scattered with blazers and school bags used for competing games of football (soccer).

3. Exams, exams and more exams.
I gather they're mostly called 'tests' here and are more often than not of the multiple-choice, verbal-reasoning variety. If you're a Brit, the months of May and June remind you of sitting at lonely desks watching a clock tick increasingly slowly while teachers pace up and down beside you. The last weeks of August remind you of getting results and everyone on the local news that night doing better than you.
Most Brits I know date themselves by what was happening when they took their GCSEs or 'A' levels. Eg, "The year I took my GCSEs was the coldest June on record and so-and-so's hands froze up and she had to go to the bog with a teacher and put them under a hot tap." Or, "When I was supposed to be revising for my 'A' levels, we went out one night and my friend Dave met Damon Albarn in a pub and twatted him one for writing that irritating 'Country House' song."

4. Rainy Day Playtime - your infancy was filled with days on end when it rained and you couldn't go outside. On the days you got to go outside, you spent most of your time looking up and worrying about whether it was going to start raining harder any minute. When you were stuck inside, teachers made you do extra work. Or clean the 'common areas' or something equally soul-destroying and stiff-upper-lip building.

5. Half term holidays. Oh how I wish I still had half terms. I think they're a brilliant concept for kids - less so, I gather for parents, who are unable to travel or do anything with the kids because the clever travel companies jack up the prices of everything over half term.... But I could really do with a half term now. I still think of half term when it gets to the end of October, in particular. I can't imagine that Spring Break, for all that it features in so many high school movies, can really be all that... I'd rather have a half term holiday any day.

So for me, British schools helped make me British - and I didn't even once have to pledge my allegiance to the Union Jack. But beyond pledging to a flag or singing an anthem - were there any features of your school life that make you feel like you have something that identifies you, or connects you, with your fellow countrymen?