In Icelandic, you are, I suppose, more likely to refer to a single person and their family, than to the family without naming any single person as well. Random example from the web: "Fjölskylda Einars Darra Óskarssonar heimsótti mig í forsætisráðuneytið í dag." ("The family of Einar Darri, son of Óskar, visited me in the office today.")
Those "last names" of Icelandic aren't family names at all. So it doesn't matter that they differ within the family; the first names differ, too. Those "last names" are generally patronymics, i.e., names derived from the given name of one's father.
Imagine a male person with a given name A. A's son with a given name B would be called "B, A's son", while A's daughter with a given name X would be called "X, A's daughter". The difference between a patronymic and a family name becomes apparent in the next generation. Imagine that B has a son who is given name C. Now, C's full name will be "C, B's son".
Finally, imagine A, B, X and C holding an extended family council. They don't use a family name, and there's no single name of any kind that would be present in everybody's full name. Will that detail stop you from calling that congregation of people "A's family" ("fjölskylda A")? No, it won't stop you, like the fact that X's (or even B's) counterpart in another Western culture might already be married and bear a family name different from the one they were born with into A's family, will not stop you either.
Having family names in place isn't required for references to families to be possible.