> but they can also (AFAIK in many states/countries) give up the child for adoption (in which case they also don't need to pay child support, as the taxpayers/adoptive parents cover the child's costs).
In most jurisdictions, giving a child up for adoption legally requires the consent of both biological parents; it is not something the birth mother can do unilaterally. (Courts do have the power to override the requirement for consent of either or both parents; but, they have to give each parent the opportunity to be heard in court first.) So, when it comes to adoption, the position of the mother and the father are more parallel than you present it as.
I'm not a lawyer and only did a bit of research in this topic, but I found out about things like "baby boxes" that allow mothers to relinquish newborns anonymously.
On the other hand, this US website [1] suggests that fathers do have a right to assume the father figure (so you're right), although their rights are still lesser than a mother's ("Most states require a birth father to provide financial support to the pregnant mother, and failure to do so may result in the birth father being unable to stop an adoption.").
Although I wonder if in the case where the father assumes custody, the mother (who wanted to give up the child for abortion) would be required to pay child support or not (the website suggests "no", but isn't very explicit on this topic).
> I found out about things like "baby boxes" that allow mothers to relinquish newborns anonymously.
Laws permitting that sort of thing only exist in certain jurisdictions. All 50 US states now have laws allowing parents to anonymously abandon infants to certain "safe places" (such as hospitals, fire stations, etc)–usually it is limited to the first 30 days or so of life. However, many of those laws are actually written so that either birth parent can legally do this, although in practice it would mainly be the mother. Also, if the other parent is aware this has happened, they can intervene and object to the process. By contrast, in England and Wales, for either parent to do this is a criminal offence (although they are unlikely to be prosecuted in practice)
> Although I wonder if in the case where the father assumes custody, the mother (who wanted to give up the child
In general, most child support statutes are written in a gender-neutral way. The custodial parent can claim financial support from the non-custodial parent, irrespective of their genders. (Or even, in some cases of joint and equal custody, one custodial parent may be able to claim support from the other.) So, even though in practice it mostly involves non-custodial fathers paying money to custodial mothers, if the father has custody, the mother can be made to pay child support. Even though it is less common, it can and does happen [1]
In one case of which I am personally aware, here in Australia, the mother had full custody of two boys. But, she wasn't sending the boys to school (among other issues); so their father went to court, and got full custody of them awarded to him instead. Legally he has the right to claim child support from her, although it is unlikely in practice he's going to get any significant money out of her (she's unemployed and of no fixed address, etc.)
In most jurisdictions, giving a child up for adoption legally requires the consent of both biological parents; it is not something the birth mother can do unilaterally. (Courts do have the power to override the requirement for consent of either or both parents; but, they have to give each parent the opportunity to be heard in court first.) So, when it comes to adoption, the position of the mother and the father are more parallel than you present it as.