No, it would be fraud and illegal

No. The birth certificate is a legal document. If you don't put the correct name(s) on it you would be committing fraud.

what sort of question is this? Whether you live in Texas, australia or even on the moon it is illegal to put a random persons name on a birth certificate. I can not understand why you would want to do this

I am assuming perhaps they were trying to get around adoption and use another women's name , just may guess.

I am not sure what OP is asking.

Can a birth mother put another name on the BC as the mother?

Strictly speaking, Vital Statistics puts the names on the BC. The parents simply report the information to them.

But for the birth mother to "put" another name on the BC as mother would be absurd. Witnesses such as attending physician wouldn't sign such a bogus report.

This seems to be the report used by Texas vital statistics. It's strange, because there doesn't seem to be any place for the informant (mother) to sign, or even identify herself.