That last line of the story reads as follows: “Our name - Zildjian - means son of a cymbal-maker. So that's what we do - that's our name.”
Well, sort of.
Zildjian is their maiden name, which they may or may not have kept after marriage.
But presumably their kids all have the father’s name, so--and here’s my favourite part--once the next generation takes over, Zildjian will be run by people with some other name, and Sabian will be run by people named Zildjian! I love it!