Do all norms agree on which vector is the longest? Given two norms $\|\cdot\|_a$ and $\|\cdot\|_b$ on a finite-dimensional vector space, could it be that there exist vectors $\vec{v}_1,\vec{v}_2$ such that
$$\|\vec{v}_1\|_a>\|\vec{v}_2\|_a$$
and
$$\|\vec{v}_1\|_b<\|\vec{v}_2\|_b$$
i.e. that the norms "disagree" on which vector is the longest?
Maybe this is related to norm equivalence, but I don't quite see how to get here from there.

Yes. Otherwise the norms would only be multiples of each other.
$$\|(30,40)\|_2=50 > 45 =\|(0,45)\|_2$$
$$\|(30,40)\|_\infty=40 < 45 =\|(0,45)\|_\infty$$
