There can be many reasons why Moria is named Kozuki
>His father was a Kozuki who left Wano, married an Oni woman in west blue, and came back
>The Wanoese got confused because his name Gekko is the kanji for Kozuki backwards
>The Wanoese decided to call him Kozuki because he was their hero and they needed a bright light under Kaido's tyranny
Moria as a kid wore western clothes seems to have been very cared for so he probably had his parents and wasn't treated poorly by the Wanoese people