a custom encountered both in Poland
Polish Americans can do all they want, I would not care less. But this tradition sounds fake to me. And the reason is this:
From: "Birthday Traditions from Around the World", coolest-kid-birthday-parties.com/birthday-traditions.html
Chinese Birthday Traditions - When a Chinese child turns one year old, it is a very important event
So which part of Poland did you say observe this custom? Serious sources, please.