What does "piecewise definition ... is not a characteristic of the function itself" mean when referring to piecewise-defined functions?

I was reading the Wikipedia page on piecewise-defined functions when, in the introduction, it states,

Piecewise definition is actually a way of expressing the function, rather than a characteristic of the function itself.

which I don't understand. If the only method of expressing a function is as a piecewise-defined function, doesn't this make it a characteristic of the function? In other words, isn't being "piecewise" a characteristic of a "piecewise-defined" function?

Well, one can define the identity function $id:\Bbb R\rightarrow\Bbb R$ as a piecewise function such as $id(x)=x$ for $x\geq 0$ and $id(x)=x$ for $x<0$. But this is not a characteristic of the function.