Why is it popular to teach modulus via the example of mod 12 and analogue clocks rather than rectangles or tables that have a finite number of columns in each row, and infinitely many rows?
It's natural to identify an origin row that has the number one in the far left-hand column of the origin row, for people who write from left to right. Then it can be indicated that the table extends infinitely far upward by filling in the blanks with zero and negative values, and the table also extends infinitely many rows down, just as for a Cartesian coordinate system. Thus, every integer appears somewhere in the table.
Focus on the clock analogy tends to restrict attention to mod 12, but of course it does also suggest going beyond integer values altogether.
We could define mod $2\pi$ as follows:
t is congruent to u (mod $2\pi$) if and only if $\sin(t) = \sin(u)$ and $\cos(t) = \cos(u)$.
Then we can generalize beyond $2\pi$ to an arbitrary positive real number $m$ by selecting a radius $r$ so that $2\pi r = m$.
Is that why the clock analogy is used?