Per Nystul's Magic Aura (PHB page 263), emphasis mine,
You place an illusion on a creature or an object you touch so that divination spells reveal false information about it. ... You can ... change the object’s magical aura so that it appears to belong to a specific school of magic that you choose.
Per Identify (PHB page 252), emphasis mine,
1st-level divination (ritual) ... If it is a magic item or some other magic-imbued object, you learn its properties and how to use them, whether it requires attunement to use, and how many charges it has, if any. You learn whether any spells are affecting the item and what they are.
The caster of Identify would see the false information that the caster of Nystul's Magic Aura placed on the object. The caster of Identify would also know that Nystul's Magic Aura was cast on the object, and would be skeptical of any other information learned from casting Identify.
In addition, Nystul's Magic Aura cannot prevent Identify from detecting that an object is currently affected by Nystul's Magic Aura. Identify specifically lets its caster know if the target is affected by a spell. Nystul's Magic Aura does not specifically give the ability to bypass this ability of Identify. The closest one could get is making the object seem non-magical in hopes that the caster of Identify never thinks to cast Identify in the first place
Per your second question and my italicized emphasis of Nystul's Magic Aura, any divination spell that can detect a spell's school of magic would fit the criteria, as this spell seems to imply that the school of magic causes the magical aura to appear or behave a certain way.