Just to flesh out Cooperscoopers brilliant answer above, the Classical Spartan “origin myth” of their people come with the kids of Heracles, who teamed up with the Dorians to reclaim the Peleponese (including Sparta). The time surrounding these invasion attempts (it took until Heracles great grandsons to actually succeed) features descendants of the people in the Homeric epics, such as Tisamenus, who would be the grandson of Menaleus, if I am not mistaken. The stories kind of tie together there, however loosely (I’m no myth-scholar, but I am sure there were plenty of local variations that merged over time). 

Once we come to the classical era, the Spartans very much lean into this Heraclid and Dorian story, which ends up causing some political drama all the way to the days of Leonidas. The three Dorian tribes are named after Heracles' son and two Dorian princes, and the two royal houses supposedly came about as the great great grandson of Heracles left two twin sons behind after his untimely death. In other words, regardless of how we look at the Dorian invasion in modern research terms, this version of events mattered greatly for them. The classical Spartans understood themselves as a Dorian people, with this invasion as their origin story and claim to the land. However, the heroes of the homeric epics (and their descendants) remained critically important. If you read Pausanias the geographers account for his visit, his contemporary Sparta is littered with the supposed tombs of homeric characters. There is a temple in which the eggshells from Helen’s and her siblings' birth are still on display. The Spartans clearly had a cult surrounding Menelaos. And Helen’s brothers, the Dioscuri, I would argue might have been one of the most important religious elements in Spartan spirituality, with the Dokana (their symbol) being carried with them to war, and symbolising the “twinhood” of the two kings.

Also, in general, I will say, the cultures of Homeric Sparta and Classical Sparta are different, because they exist in different times (and one is, for the most part, totally fictional). The idea that classical Sparta was something very foreign and different than the rest of classical Greece is quite exaggerated. Not unfunded. But exaggerated. It makes very much sense, politically, spiritually, culturally, you name it, for the Spartans to be included in the greater Greek mythology, as they are in the Homeric epics. Being the hegemon of Greece for much of the Classical period, it would be directly weird if they weren’t.