I'm not clear about the customer in Hongkong. But here in China mainland, indeed some children are named after his elder brother or sister. I.e. if the oldest son is named Jie, then the second son may be named second jie and the 3rd son third Jie... Very interesting!
But the second son of the second son has nothing to do with his uncle(the oldest brother of his father), he may be named after his own elder brother, but not his uncle. Actually, it is irreverent to have a name related with one's superiors in china mainland.