or does your name become public? if I never mentioned my name on my website or stores.
It depends upon your location. In the USA, Business licenses are public record. Property taxes are public record. Government agencies will provide certain information in the event of consumer complaints. Sellers in the EU are required to put physical contact information on their web site.
From some countries, the data can be hidden, and any consumers stupid enough to buy things often can get cheated. Only fools buy things without having direct contact ability to a seller. It’s a big red flag when contacts are email addresses or only through the web site.
Most web sites don’t want their web site copied and purposely list copyright ownership. Once doing that, it can be traced easily.