It's not really a fractal in the sense that a fractal is a mathematical object, and a coastline lives in the real world. As your ruler gets smaller, it also becomes impossible to define what even is the coastline anymore - is it the wetted sandline? On a sufficiently small scale, where is the interface between land and sea?

Thus at sufficiently small scale it does break down, but the fractal nature over a very broad range of scales is clear; I feel like the Wikipedia article is tripped into pedantry at that point :)

On a sufficiently small scale, I guess you would have to start worrying about quantum effects. So, you would still have trouble measuring the coastline maybe not a paradox anymore but still quite a pain.