Not in 5e, but they did in previous editions
5e's rules don't seem to make any general connections between magical items and schools of magic. An active spell clearly belongs to a particular school, as all spells do; but magic item descriptions don't reference schools of magic. Detect Magic tells you the magical school associated with some magical aura, "if any", so it's evident that a magical effect doesn't have to belong to a particular school.
It would often be reasonable to say that a magic item which produces a particular spell effect, such as a wand or scroll, bears the corresponding school of magic of the spell, but there is no rule that states this is the case.
However, in earlier editions of D&D such as 3.5e, magical items absolutely were associated with the schools of magic. As described in the 3.5e SRD, Detect Magic tells you what schools of magic an item corresponds to, and every description of a magical item includes a note on the strength and schools (many items belonging to more than one school!) of its magical aura. DMs who are apt to describe magical items with particular schools of magic may well be familiar with previous editions and extrapolating to 5th edition.
In any case, it is of course up to the DM if they want magical item auras in their game to indicate a particular school of magic or not. I would prefer to do so in my games, since it's a good way of suggesting the likely function of an item without being explicit about it, and gives the impression of consistent rules of magic. I would consequently also rule that mechanical interactions based on schools of magic could apply to magical items, such as in the linked example of the Robe of Eyes, and so Nondetection should protect against the magical senses of the robe. That's definitely my personal house ruling, though.