Does $\pi \ | \ 2 \pi$ Does $\pi$ divide $2 \pi?$
Clearly $\frac{2 \pi}{\pi}=2$ and 2 is an integer, so it would seem to make sense to say that $\pi \ | \ 2 \pi$.
Does it make sense to write, for example, $$\pi \ | \ x \implies \sin(x)=0?$$

Yes, it makes sense. It is not widely used, but it is clear and useful.
