Marriage does not have to be religious, and it's not exclusively religious in origin. Many millions of married yet irreligious people who had zero church involvement would take issue with that assertion.

I don't see the point in doing this even if it was. It's just semantics. We'd still need a legal shorthand for all the rights and responsibilities currently attached to marriage, as people would still want that. Then it's just marriage by another name.

Also, I'm not sure any of these countries "force" any church to recognize a marriage they don't agree with. That wouldn't change, since I'm sure different churches would still disagree on which marriages count.