I do not think it should be true, as if your green card is issued by a Federal government and shows your birthday, it should be accepted. However, I can imagine that a store may have its own policy. If a passport is accepted as a form of ID, a permanent resident card should be accepted as well.
Please understand that most store clerks are not trained to identify each and every piece of govt-issued document. Therefore, it has more to do with the clerk's lack of experience or knowledge than with ur GC. They are accustomed to Canadian IDs in Fla as a lot of Canadians come down here, but a GC is a less frequently used item. Any reason why you cannot get a Fla ID? It might make life easier.