Gin is a bit of an odd duck when it comes to spirits, since most of the flavors come from additives that are incorporated into the spirit after it has already been distilled at least once. You can read more about it here, but the big differences in the spirits are typically in the level of juniper in the flavor profile and the kind of infusion process that is used.
Despite the name, a London Dry Gin doesn’t actually have to be made in London. Instead it’s a marker of a specific style, that of a traditional gin where juniper is the predominant and forward component of the flavor profile. These gins can be made anywhere in the world as long as the juniper remains large and in charge.