This is a good question and I can see why you might be confused on this topic. In the Odyssey by Homer, there is no record of the birth place of Odysseus. For this reason, we can only make an educated guess.
We know that Odysseus was the king of Ithaca. Moreover, we know that he was the heredetary king of Ithaca. This is why his son would be the one who will succeed him. For these reasons, it is likely that Odysseus was born in Ithaca and grew up there as well.
Two more points support this thesis. First, his dog recognizes him. Second, his old nurse recongizes him. They probably knew him well as they watched him grow up in Ithaca.