In a question about the Robe of Eyes detecting someone using an amulet of nondetection, the consensus was that the magical effects of the Robe of Eyes was not considered divination magic. However, it got me thinking about Detect Magic and its ability to determine the school that a magical effect comes from.
A lot of DMs will say "you get a very strong feeling of abjuration magic from this amulet of defense". Is this accurate and items have associated magic schools?
5e's rules don't seem to make any explicit connections between magical items and schools of magic. An active spell effect 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, and by the rules as written, using detect magic on a magical item only tells you that the object is magical, nothing else.
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.
As well as this history, although 5e's text does not explicitly assign schools to magical items, digging into the art can reveal implicit connections. As explored in this question, DnDBeyond uses a particular set of symbols to represent the schools of magic and that symbology is also used in some of the official art of items. For instance, the artwork for the aforementioned amulet of proof against detection and location clearly shows the abjuration symbol over a closed eye; the clasp on the cloak of protection or mantle of spell resistance is likewise an abjuration symbol; the defender sword forms the shape of the abjuration symbol; the eyes of charming have a (version of) the enchantment symbol in their lenses; the iron flask has symbols of abjuration and conjuration embossed on it; and so on. So at least one person on the art team was deliberately making these connections in the item art.
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. If they do, they will likely have to refer to the PHB's description of the individual schools of magic and their own judgement to identify where any given item belongs in that categorisation, though resources from 3/3.5e which describe the same or similar items would be a useful reference too. At the very least, it would seem reasonable to say that a magic item which directly casts or reproduces a particular spell effect, such as a wand or scroll, bears the corresponding school of magic of that spell.
Personally, I would prefer to include this 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. But that is only my personal house ruling, though; it is not supported by 5e's text at all.
