Metronym vs Matronym

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.

From Wictionary

metronymic - English. Alternative forms · matronym. Etymology. From Ancient Greek μήτηρ (mḗtēr, “mother”) + ὄνυμα (ónuma, “name”).

thefreedictionary.com/Metronym

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.

Merriam Webster

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.

In Ancient Greek, the word meter (μήτηρ) means "mother". It has "e" in the first syllable because the Attic dialect of Ancient Greek—the standard dialect used as the source for loans into Latin and English—had a sound change that turned long a sounds into long e sounds. Certain other dialects of Greek, as well as Latin, did not have that sound change, and so used a long a sound in the first syllable of the word for mother.

The modern use of "Metro" to refer to things related to cities is shortened from metropolis (Ancient Greek μητρόπολις) which is a compound of the Greek words meter "mother" and polis "city".

So in terms of how it works in Greek, "metro-polis" = "mother city" and "metr-onym" = "mother name".

Since English speakers don't necessarily recognize "metr-" as meaning "mother", some speakers may prefer to use "matr-onym". "Matr(i)-" with the meaning "mother" is used in a fair number of English words, such as matrilineal, matriarch(y), matricide.

I haven't encountered the use of "metr-onym" with the sense "name related to cities," so I'm not sure how common it might be. Maybe Butcher intended to coin a new word that coincidentally happened to have the form of a prexisting word with a different meaning. I think the use of "metronym" in the quoted passage from Captain's Fury was probably based on the meaning of English "metro(-)" rather than on the meaning of the Ancient Greek root.