I recon there is more of the whole reusing one child's middle name for a subsequent child's first than we realise.
It happened in my extended family. Like Louis, the reused name was a family name. I guess they gave it as a middle name in case they didn't have another son, and then the other son came along after all.
In George's case, George was presumably to please the queen, I've seen it suggested that Alexander was Kate's choice, and Louis would have been for Philip and Charles.