Schengen is absolutely an EU achievement, although for some time it developed separately; it is now absorbed into EU law. Note that every Schengen country outside of the EU is a member of EFTA and to be in Schengen has to sign an association treaty with the EU.

Yes. Though it's part of the same broader 'European project', so it's permissible for people to be a bit lax when informally discussing these things.

Where have I said that on my comment?