"I think the father's rights have some restrictions in the US"

There's always catch somewhere, and there have been a lot of jingo-istic gotchas in the past (eg, a US kid born overseas used to have to visit USA for some time before the 18th birthday to retain citizenship or something like that (but the USA wouldn't pay for the airline ticket, only force you to do it). Another was that US citizens couldn't have dual citizenship. dumbass red-state/redneck sh!t), but most them have been removed. I think you still lose US citizenship for joining a foreign country's military/national service or police force, IIRC.