I'm not sure what the issues are in the Ukraine, but if it was part of the Austro-Hungarian empire at the time, often Jews would marry according to Jewish tradition (with a ketubah) and not civilly. As a result, when children were born, they would not be registered under the father's name; they were noted as being unehelich or illegitimate and recorded with the mother's maiden name. So if she happened to have been born a KOHEN/KOHAN/KOHN etc., the child would bear that name.
Some families did later undergo civil marriages and the names might then have reverted to the father's name, or not.
Beverly Hills, CA
Researching: GOLDMAN, STEINER, LANGER, GLUECKSMAN, STOTTER in various parts of Galicia, Poland
(Nowy Targ, Nowy Sanz, Wachsmund, Dembno, Lapuszna, Krakow, Ochotnica) who migrated into Kezmarok or
nearby towns in northern Slovakia and Czech Republic (i.e., those who lived/had businesses in Moravska Ostrava);
GOLDSTEIN in Sena or Szina, Szkaros and Kosice, Slovakia; Tolcsva and Tokaj, Hungary.