Yes, "through" is commonly used informally to mean "finished" so I suspect it probably means that Jane has just started her summer holidays (AE = vacation).
Had I been Jane, I think I'd have said (in BE). "We've just broken up for the summer [holidays]".
Yes - thanks (to the many) for pointing out that "I'm through for the summer" means I've finished school and am now free "for the summer" to do what I want.
I see no way to arrive at this interpretation.
But not everyone's school year ends on the same day, or even in the same week.Saying "I'm through for the summer" would be pointless in normal circumstances. It's not summer yet until after you're done and in that sense everybody is through for the summer when the school year ends.
Nowadays, universities have classes year round. Summer classes are very common albeit smaller and shorter but still abundant. At my university we have three small summer semesters in May, June, and July.My first thought was different. If I heard that sentence I would assume Jane had been taking extra classes in the summer and that she had now completed those classes. She's now through for the summer. She did have summer classes, which Jack knew and was why he was asking her the question, but now she is finished and has no more school obligations until fall.
In the U.S., the school year everywhere that I know about finishes well before the official start of summer in the third week of June. Unofficially, summer starts when the school year ends, which is about the time the weather starts getting seriously warm.
Saying "I'm through for the summer" would be pointless in normal circumstances. It's not summer yet until after you're done and in that sense everybody is through for the summer when the school year ends. But in the case of individual students taking summer classes, it would be a very meaningful sentence. I think where I lived summer school lasted three or four weeks, and the summer was longer than that. "I'm through for the summer" would describe that time after the last summer class but before the start of the new school year.