I know that there is also the variant 放工 but I don't think this is typical usage.
The use of 下 is so common, but yet in the instance of a school we use 放 instead.
Let me do some speculation.
Well, since we have 上班 & 下班, why not 上学 & 下学?
I suppose it is 放学 because 放 also has the meaning of "being let out or let off", or "released from some physical restrictions, like the school compound" from which students could not "get out" from during school hours?
So, 放学 is meant to depict the actual physical situation of students being 放, i.e. "let off or out" of the school compound?
Workers on the other hand are not so functionally restricted during their hours of work and so 上班 & 下班?
放工 perhaps have the same meaning as 放学, meaning workers being "let off or released" from their work. 放工 usually, though not always, refers to menial workers.
Like I say, I am just speculating.
上 and 下 represent a plain cycle that carries not much excitement/change before and after, or in between, therefore, it is best to associate the words with the everyday events - 上班 and 下班.
However, for most school kids, 上學 equals a day of torture in a confined environment -the school, and he/she couldn't wait to be freed/released(放) from it to enjoy the after-school hours and funs. Thus, 放學 (free from school) is more appropriate than 下學, which has no expectation or excitement attached to it.
Note, similar to 上班 and 下班, the words for start and end of a single class are 上課 and 下課, 放課 is not used, because the kids are not really/completely let free after the recess.