When "the school holidays" is understood to be a span of time, during can be used and on, which requires a firm landing place, a spot, not a duration, cannot.
Bears hibernate during the winter.
Bears hibernate on the winter. ungrammatical
In temporal contexts, the object of preposition on must be a single place in time (or a set of single places in time).
She visits her grandmother on Tuesday.
She visits her grandmother on Tuesdays.
She visits her grandmother on the summer. ungrammatical
Tuesday refers there to a particular location in the week, not to a duration of time.
The train departs on the hour.
There hour is understood not as a duration or span of sixty minutes but as a point in the day, a point which we reach every sixty minutes, like inch-markers on a ruler. Just as we can understand an inch either as a span or as a mark on a ruler or yardstick, we can understand hour as either a duration or as a point in time along the length of the day.
The flip side of the coin is that during cannot be used with single places in time, since it needs a duration, not a point in time:
She visits her grandmother during Tuesday. ungrammatical
She visits her grandmother during lunch-hour on Tuesdays.
But it is also possible to understand lunch-hour as time-being-spent-in-a-particular-manner, analogous to a journey or a vacation, which allow use of either during or on:
She visits her grandmother on her lunch-hour on Tuesdays.
She visits her grandmother during her lunch-hour on Tuesdays.
What did you do during your vacation? in the time you were away
What did you do on your vacation? on the occasion of being away
So, which preposition to choose depends on whether the noun phrase in question is generally understood to be a point in time, a duration of time, or an occasion that occupies time and thus admits either sense.