dnd 5e – Do magical items have a magic school?

Not in 5e, but in previous editions

The rules of 5e seem to make no general connections between magical objects and magic schools. An active spell clearly belongs to a particular school, as is the case with all spells. But descriptions of magical objects do not refer to magical schools. "Detect Magic" tells the magical school associated with a magical aura, "if anything". So it's obvious that a magical effect does not have to belong to a particular school.

It is often reasonable to say that a magical item that creates a certain spell effect, such as a For example, a wand or scroll carrying the appropriate spell school of spells, but there is no rule in which this is the case.

However, in previous editions of D & D like 3.5e, magical items have been absolutely associated with the magic schools. As described in 3.5e SRD, Recognize magic tells you which magic schools an item belongs to, and every description of a magical item contains an indication of the strength and the schools (many items belonging to more than one school!) of its magical aura. DMs who are able to describe magical items with certain magic schools may be familiar with previous editions and can extrapolate to the fifth edition.

In any case, of course, it's up to the DM to see if magical Itemauras are supposed to display a particular school of magic in the game or not. I would rather do that in my games, as this is a good way to suggest the likely function of an item without being explicit about it, and gives the impression of consistent rules of magic. I would therefore also fear that mechanical interactions based on magic schools could be applied to magical objects, as in the linked example of Robe of the eyes, and so No detection should protect from the magical senses of the robe. This is definitely my personal house decision.