Does the length of a river depend on the scale of measurements?
The same problem applies to all shapes that are complex and this includes coastlines, rivers and the perimeters of leaves. Most shapes derived by nature will have this issue. The unit of scale matters.

Only shapes that are composed of straight line segments and can be easily measured as such do not have this problem: triangles, rectangles, hexagons, etc. Such shapes are mathematical or created by humans.