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.
Wish can duplicate any other spell, even one not on your list
You've quoted the important part yourself:
The basic use of this spell is to duplicate any other spell of 8th level or lower.
There are no qualifications or restrictions to this statement beyond what is said here. As long as it is 8th level or lower you can call any spell.
If there were any further restrictions it would say so in the spell description.
Best Answer
There are some other benefits that aren't as obvious.
First of all, you gain access to many spells that are either not known/prepared or appear on other class spell lists.
Possibly more potent, however, is the removal of expensive material components and long casting times. Some spell like resurrection, simulacrum, or temple of the gods have much more powerful effects without their material components and casting times. You simply have to speak their effect into existence.