The spell's school is whatever the duplicated spell's school normally is
The basic use of this spell is to duplicate any other spell...
Duplicate here mean to "make or be an exact copy of". Note that the wording says the entire spell is duplicated, not just its effects. An exact copy of a spell means that it is identical in every aspect, which includes school.
If wish was intended to only take some of the characteristics of the spell and not others it would say so. In fact, it does say so, but only for two specific things.
Wish specifically indicates things that are not duplicated from the spell
You don't need to meet any requirements in that spell, including costly components.
Wish tells us that only one thing is different casting a spell using wish versus casting it normally: requirements of the spell do not need to be met.
...The spell simply takes effect.
Also, it is implied by the wording "the spell simply takes effect" that wish also overrides the casting time of the duplicated spell.
Since no other exceptions are noted the spell should be treated exactly as if the spell had been cast normally including duration, range, school, etc.
Wish only lasts for an instant
Wish is an instantaneous spell which means no matter what it is gone the instant after you cast it. But say you used wish to cast a spell, say geas, on a creature using wish. The instant after wish is cast the only effects remaining are going to be of the spell it is duplicating not wish itself.
Wish conjures a duplicate of another spell
Essentially, it seems that using wish to duplicate another spell would result in the casting of a conjuration spell with an instantaneous duration which then results in another spell taking effect with all the details that that spell has (including school). Wish conjures the other spell into existence and then disappears leaving only the conjured spell's effects.
You will still get stress
Jeremy Crawford has clarified exactly this in this tweet:
Wish spell: (1) Duplicated a spell of 8th level or lower? No stress. (2) Did anything else with wish? Stress. #DnD
A normal reading of the rules you already quote also specify "any effect other than duplicating a spell" so I think it is very clear that this is the case
Best Answer
No. Wish uses a 9th level slot for any casting, and one of its canonical uses is to duplicate any spell (of 8th level or lower). Consider the case of a 19th level Wizard who, for one reason or another, has expended all of their 7th and 8th level slots, and further, doesn't actually have Mordenkainen's magnificent mansion prepared -- but wants to cast it in order to take a safe long rest, recover their slots, and incidentally make a couple small changes to their prepared list.
They have wish at their disposal, and a 9th level slot -- if casting wish required also having a slot of the level of the spell it's duplicating, we'd have a case of the most powerful and versatile spell there is, being unable to do one of the things it's designed to do.