The premise that Cyprus lacks a national anthem is incorrect. The Republic of Cyprus adopted "Ýmnos eis tīn Eleutherían" ("Hymn to Liberty") as its national anthem in 1966, six years after gaining independence from Britain in 1960.
It is true that Cyprus and Greece share the same anthem—a unique situation reflecting their deep historical, cultural, and linguistic ties (the majority Greek Cypriot population identifies ethnically as Greek). The comparison to Liechtenstein (which uses the melody of "God Save the King/Queen" for its anthem "Oben am jungen Rhein") and the UK is apt: multiple sovereign states can share a melody or even lyrics. This does not negate Cyprus's having its own officially adopted anthem.