While Icelandic shares deep historical roots with the Scandinavian languages, it is not classified as a Scandinavian language in the strict linguistic or geopolitical sense.
Icelandic belongs to the North Germanic branch of the Indo-European family, alongside Norwegian, Swedish, Danish, and Faroese. The term "Scandinavian languages" typically refers only to the mutually intelligible trio: Danish, Norwegian, and Swedish.
Icelandic is the most conservative of the North Germanic languages. Due to centuries of isolation, it has preserved complex grammar (four cases, three genders) and vocabulary from Old Norse far more faithfully than its mainland cousins. Modern Icelanders can still read 13th-century sagas with relative ease.
Think of it this way: Icelandic is a North Germanic language (like the Scandinavian ones), but it is Insular Nordic, distinct from the Continental Scandinavian group.