Learn English – A word that describes giving its name to something else: iron is … for ferromagnetism

A set phrase, a fixed expression, an idiomatic expression: all these would seem excellent terms to describe what you mean. They all imply that the phrase means more than the sum of its parts, as one would ordinarily expect, and is often found in this exact form. These terms serve their purpose quite well; I doubt whether anything better exists.

If you want to go with one word, it might be censorious: (or censoriousness for the tendency/urge)

having or showing a tendency to criticize someone or something severely : very critical

The details are more related to psychology but this urge might come from chronic criticism (or having a critical mind).

Psychologytoday mentions that these people are usually self-critical also and this transforms into chronic criticism of others.

Critical people tend to be highly self-critical. As hard as they might be on others, they are usually harder on themselves. They were often criticized as children, at least implicitly - the message was clear that, in important ways, they weren't quite good enough. Self-critical patterns tend to form in early childhood and by late adolescence mutate into chronic criticism of others.

Also mentioned is the characteristics of a chronic critic:

Feels compelled to share an (unfavorable) opinion on just about everything. Has never seen an artistic masterpiece or a preschool graduation that couldn't use a bit of tweaking. Prone to eye rolling, smirking, and/or slow-motion head shaking, as well as a tone best described as oozing with condescension.