If it were the case then no american would be an american. They all would be british!
Well, certainly America (the 13 colonies) was British at one time.
By the way I have no ancestry from British people or their descendants.
Just being born in a particular place has no necessary connection to one's upbringing, values and loyalties. That's the point.
I suppose it could be said the same thing is true of who one's parents are, yet the very fact that someone's parents come from a particular land binds a person to that land more than merely being born in a land.
As a Chinese woman once told me, just because someone becomes an American citizen they don't become Americans. A Chinese person with American citizenship is still Chinese.