Argthtjtr

In Jane Eyre by Char­lotte Brontë I read:

“Burns” (such it seems was her name: the girls here were all called
by their sur­names, as boys are else­where)...

So my ques­tion is: were there (or are there) dif­fer­ent rules for ad­dress­ing
boys and girls in English schools? And if so, how were the girls usu­ally
called if not by their sur­name?

British schoolboys were traditionally addressed by their surnames (if two brothers attended the same school they would be 'Jones major' and 'Jones minor'). I went to a girls' school in the 1960s, but I attended some classes at a mixed school where boys were called by surnames and girls by their first names. I don't know if this custom is still observed; I guess probably not, and it's first names all round.

Jane Eyre evidently found it surprising that the girls at Lowood School were not called by their Christian names. I believe Charlotte Brontë based the school on one she attended in real life.

(Disclaimer - There are already some great answers and I know my answer is anecdotal and therefore low quality, but I think it has some information and context to impart that many native british-english speakers would take for granted and therefore not think to explain)

Just adding my two cents: This is the norm in Australia too. Although in public schools (government - not private) boys and girls are called by their first names, boys and men are still often referred too by their surnames, especially in sports, or male-heavy activities/locations such as at the pub. It's kind of a 'blokey', camaraderie-inducing thing to do - funnily enough, since you might think it would be more formal, and therefore impersonal and distancing but it is in fact the opposite. It doesn't depersonalise because it's essentially a practical thing: in a class with 15 students named John, calling them by their last name actually individualises them.

(I can imagine there are some female sports coaches that also refer to players by surname, but outside of sport it is practically unheard of. It wouldn't be insulting, necessarily, just very jarring and unexpected (and in the case of a common surname, it would not even cross a woman's mind that she was being referred to / addressed))

In fact I was very chuffed, and really felt 'part of the team' when my first boss (quite a blokey, sports-loving guy) called me by my surname, like he did all the other employees (who were all men). I work in a male dominated field (computing) and am a woman, and this (calling the men by their surname but not the women) is the kind of subtle thing, often stemming from nothing but politeness, that can make you feel like an outsider. So I was very pleased and smiled inwardly every time he used my surname. It has never happened again before or since, in my life - professionally or personally.

A very exaggerated patriarchal reading is that men are the extensions/representative of their family line, and the things they do, good or bad, reflect on more than just themselves. As well as inheriting their name, they will also inherit reputation, property and perhaps even titles. Using their surname (their 'father's name') reminds them of this weight, and also privilege, encouraging them to act more maturely.

Rather than diminishing their power, it instead represents that they are expected to effect change on the world around them, and their name will be attached to that, therefore implying (thus granting) more agency and power.

Whereas women just need to look pretty in a sitting room and won't ever be known outside their own home, let alone change or contribute to the wider world, so you can basically call them anything; and they only have their father's name until they are married, so what's the point in emphasising it?

In a nutshell: mens' names indicated representation, womens' names indicated ownership.

So a school that calls the girls by their surnames would possibly have more of an ethos of raising well rounded, educated, capable citizens of the world, rather than preparing, polishing, and baby-sitting obedient and inoffensive girls until they're married off.