When I look up the definition for Metronym or Metronymic, the definition says that it's an alternative form of Matronym, which is a name taken from a person's mother.
metronymic - English. Alternative forms · matronym. Etymology. From Ancient Greek μήτηρ (mḗtēr, “mother”) + ὄνυμα (ónuma, “name”).
mat·ro·nym·ic (măt′rə-nĭm′ĭk) also me·tro·nym·ic (mē′trə-, mĕt′rə-) adj. Of, relating to, or derived from the name of one's mother or maternal ancestor. n. A name so derived.
However, Metro is related to a French word for a major city or metropolitan area, so Metronym would seem to be a name taken from a person's place of birth, or a family seat like Leonardo Da Vinci or Catherine of Aragon.
metro - Noun. French métro, short for (chemin de fer) métropolitain metropolitan railroad
however, it's also related (at least phonetically) to Metropolis or Metropolitan which Webster defines thusly:
metropolitan - noun. 1. the primate of an ecclesiastical province 2. one who lives in a metropolis or displays metropolitan manners or customs
metropolitan - adjective Definition of metropolitan (Entry 2 of 2) 1 : of or constituting a metropolitan or his see 2 : of, relating to, or characteristic of a metropolis and sometimes including its suburbs 3 : of, relating to, or constituting a mother country as distinguished from a colony
History and Etymology for metropolitanz: Adjective - Middle English, from Late Latin metropolitanus of the see of a metropolitan, from metropolita, noun, metropolitan, from Late Greek mētropolitēs, from mētropolis see of a metropolitan, from Greek, capital
I've seen the word used in the latter sense by Jim Butcher in Captain's Fury (chapter 55) when describing those who belonged to the common House of the city they were born in (e.g. illegitimate children of nobles) such as Antillar Maximus and Phrygiar Navaris.
From "Captain's Fury by Jim Butcher, Chapter 55
The name Phrygiar, like every other metronym in the Realm, was an indicator of illegitimacy. Children who were not recognized by their fathers and admitted into their Houses legally became members of the "city-house" of whichever High Lord held authority over their birthplace.
So the question is: is Butcher using the word wrong, or is this a very uncommon definition that dictionaries don't bother mentioning?
honestly, the in-context reading of the definition would seem more logical.