Culture is a shared sense of national identity which binds the people of a nation together.
National as in nation-state or national as in “nations within a nation”.
National is general a term for political boundaries, which has little to nothing to do with cultural boundaries. Just look at the way colonists divided up African states or look at the Kurds.
Moreover, your definition of culture is not functional in any way. You’re simply saying culture is a sense of identity. Culture is also a set of tools for mutual understanding, allowing people to interact without having to come to negotiated understandings in every single detail. Culture is functional and shared, but is also negotiated, contingent on language and meaning, and predicated on consensus. This means culture is malleable. It’s both reproduced and transformed. It’s the input, the medium, and the output of social interaction.
What you’re talking about is identity. And you would probably be very disappointed with a survey of people asked to describe the Canadian identity because one of the top responses tends to always involve a plurality of ethnic and cultural backgrounds allowed to co-exist in a patchwork.