In physics "order of" means "approximately" using something like a taylor series, which typically start with a constant, then move to higher polynomial terms which add smaller and smaller corrections. Similar, but different, I think...

The O stands for "Ordnung", the German word for order. So it does literally mean that, except mathematicians think that the order of f(x)=1 is the same as the order of f(x)=10^6, because "clearly" f(x)=x gets way bigger than any constant function.