Why is it important that the $p$-adic absolute value satisfy multiplicativity? I suppose my question is really "why do we require norms in general to satisfy multiplicativity?". I ask this because for the usual absolute value on $\mathbb R$, I never feel like multiplicativity plays any "key role"; compare this to for example the omnipresent triangle inequality for the usual absolute value on $\mathbb R$, and in general in any metric space.
Suppose we are learning about $p$-adic numbers for the first time, say via this nice presentation: https://math.uchicago.edu/~tghyde/Hyde%20--%20Introduction%20to%20p-adics.pdf. I get that we would like some sort of norm $|a|_n$ for $a\in \mathbb Z_n$ that is smaller for values of $a$ that have a larger number $N$ of rightmost zeroes (in base $n$), and I agree something like $c^{-N}$ is a natural choice (really given the limited menagerie of "standard elementary functions" that's pretty much our only choice, other than perhaps an inverse power function like $\frac 1N$ or $\frac 1{N^k}$), but why do we emphasize that such a norm MUST be multiplicative?
EDIT: basically, I’m looking for the easiest example of the $p$-adic norm being used in some application to prove something “interesting” (outside the abstract theory of absolute values), that requires multiplicativity; in particular this rules out Ostrowski’s theorem.
Sort-of related is Why does the p-adic norm use base p? since it is sort of related to this "motivating the $p$-adic norm" business I have going on here.

The $p$-adic norm on $\mathbb{Q}$ is a standard example of an algebraic absolute value. In particular, we are viewing $\mathbb{Q}$ as a ring (more specifically an integral domain) here, and so the only notions of absolute values that are of interest are the ones that respect the ring operations. None of the other axioms refer to ring multiplication in any way, so the requirement that $\lvert \cdot \rvert$ be multiplicative can be thought of as the simplest way to make the absolute value respect the ring multiplication.
