Birth certificate for child

If you have a child while not being married, does it show the mother and father as unregistered marriage?

I can only tell you about things I've seen, not the actual law.

An unmarried woman in my old village couldn't get the father's name on her child's certificate.

That means a bastard can't have their father's name on the family card.

An unmarried senior high student who became pregnant was kicked out of school, and I was told her family kicked her out of the house.

I was told by a local primary school principal, local schools didn't allow bastard children to register.

The law might well be different (I have no idea), or bribes might well change the situation.

Short of anything else, the above shows attitudes, thus potential difficulties.

Get married.