A couple of things I found at https://www.findlaw.com/family/paternity/legal-significance-of-paternity.html

I was wrong about birth certificates, which I should have known if I was thinking. My apologies for that.

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Contrary to come belief when a man’s name is indicated on a child’s birth certificate as the father, this doesn’t establish paternity.

On the other hand, I was right about something else:
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Under voluntarily assumed paternity the father welcomes the child into his home and open the old the child out as his own.

There are also paternal rights as well as responsibilities and I believe they’re included in the article.

Julia
A 45’s quicker than 409
Betty’s cleaning’ house for the very last time
Betty’s bein’ bad