"Lyðveldið Island" is simply "The Republic of Iceland" in Icelandic. So there is no sense in the statement that "some" locals "still" call it that today; that's just its name. As far as I know, it has always been known as "Iceland" (or "Island" in the Nordic languages) ever since the Vikings discovered it.
By the way, no the question was not referring to the shop - it was referring to the country.