Ahh now I think I understand where you are coming from.
I think the distinction here is distance vs area. A 4m perimeter is distance which is not written using squares. This is because square, or squares, denotes area.
A fence will be 4m in length.
Land or buildings covering the same area will be in square feet or square metres denoting the inner area bounded by the perimeter.
Whereas the term metres squared is usually only used in mathematics/physics formulas, which is probably why it feels like a local dialect and not an actual thing!
If it’s any consolation I routinely need to remind myself of the difference between sq metres and metres squared every time I go to review land and building valuations! Along with length only calculations for fencing.
Hope that helps!