Language is one of the primary vehicles of culture, and expressions of cultural distinctiveness. But it is not the only one.
When a language dies, the language community has been linguistically assimilated into another community (assuming the community hasn’t been genocided). That is typically associated with cultural assimilation. But not always.
As a counterexample to both: Irish culture, which is not dependent on the (now tenuous and mostly emblematic) survival of the Irish language. Yeats was no less Irish for writing in English. But Ireland is an island country, and can sustain a distinct culture more readily than, say, Sorbian.