More specifically, there is not continuity that would make it one big civilization. Later civilizations copied things like legal system from the romans and inherited language. a lot of the seeming continuity is due to renaissance copying the "classics"

It's called the "west" in opposition to the eastern block during the cold war. And yes it's a useful concept because it exists and is well defined.

No, the concept of “Western civilization” is much older than the cold War, and the West in it is more in reference to the East-West Schism (or perhaps the East/West split of the Roman Empire) than the Cold War.