Well, yes, but those born here and raised elsewhere do not have to claim citizenship, and the US cannot tax them. The US does not go hunting for people born here, who went home with their parents as infants, and never had anything further to do with the US, in order to impose Federal tax upon them.
Aside from the fact that is would not be cost-effective, the US does not recognize dual citizenship.
Regarding proof in hospitals:
I would not be shocked if in my lifetime, these 3 things happen:
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DNA tests become so cheap and quick, and surrogacy widespread enough, that hospitals routinely test the DNA of all mothers and children, and all men who wish to be named on birth certificates. Married partners of mothers, who are not biologically related to the child, will probably need to show ID and the marriage license to be named, but a password[-protected link on your phone would suffice.
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Every child born here not in a hospital will need to have the DNA tests done within a certain period of time, and at least one parent will need to show proof of citizenship to certify the child as a natural-born citizenship. If neither parent is, some sort of proof that the child was actually born in the US will need to be submitted. I am not going to speculate on what that might be.
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There will be a child born to a legally married lesbian couple or a man-woman couple, born at home, who is DNA matched to the bio-mom (or in the case of the opp-sex couple, an anonymous sperm donor has been used). Bio-mom is not a citizen, but legal partner parent is, and is legally a parent immediately from the child’s birth (or earlier, if those fetal-citizenship laws are passed) There will be a lawsuit over whether this child gains citizenship through the legal partner parent.
I don’t know what the ultimate decision will be in #3, but years of “marital presumption” might be viewed as a precedent, and the friendliness of the current administration toward SSCs, and birthright citizenship will matter.
I might not have expected these things in my lifetime if you’d asked me 20 years ago, but when I was young, like in college, I marched for gay marriage, yet did not think it would happen in my lifetime, and I did not think a black US president would happen in my lifetime. There are lots of things I could imagine, and believed would happen, but not in my lifetime.
I’m 58, and conservatively have 15 more years. I get yearly check-ups of everything, and don’t smoke, drink, nor eat much added sugar, so 20 is likely. That’s the technology and social progress of 2005 to now, adjusted logarithmically for the increasing pace.
Of course, it also assumes the US is not in its dotage, and about to descend into perturbation, disorder, delirium, and sheer catatonia.