Anyone with an American parent, who is born abroad, is considered to be an American. As it is with Ted Cruz and John McCain.
In other words, Obama's mother was American, therefore - born anywhere in the world, if that had been the case - Barack Obama would always be an American.
Neither Cruz nor McCain were harassed about where they were born, and they were born abroad. Not Obama.
Obama was born in Hawaii. The whole birtherism was spread by Trump because Trump could not stand seeing a black person as President of the US. And so did many who repeated that lie.