Also read about in loco parentis regarding this question, which is in part a legal doctrine that may allow or require a non-biological parent to exercise the legal rights and responsibilities of a biological parent if they have held themselves out as the parent. As mentioned already, it varies by location and time what “holding or being held” out to be a parent means.
My divorce agreement contains a sentence stating that I am not acting in loco parentis with respect to my ex-wife’s 14 year-old son, who has a different biological father. This means no child support owed by me, but also no parental rights or visitation.
We could both make arguments against this if we wanted. I’ve been acting as this boy’s father since he was 3 years old. However, the separation happened when he was a (fairly) independent teenager, can drive, work, take care of himself, lives with mom, and we don’t talk or see each other anyways. It would be an uphill battle to force me to support him.
Fortunately mom is of the opinion that kid’s should be allowed whatever they want, the 14 year-old doesn’t want me or my rules around, so she’s giving up child support in order to keep me out of the picture.
I don’t know how things would turn out if you were rich and had a high-price team of lawyers arguing about it though… for example the Orange one discovered today that Baron was actually fathered by Rudy. Could he get out of owing his wife for support?