[graybanana]

I think the author's point is that while you might a priori think there are lots of groups out there that might be good candidates for DH, it turns out that elliptic curves are a strictly better choice than every alternative, and the reasons for this were definitely no fully known at the time Miller and Koblitz proposed ECC.

There was a period during which there was lots of interest in using abelian varieties of higher dimension (arising as Jacobians of curves of higher genus), with dimensions g=3 and g=4 being particularly attractive because then you could work over a very computationally friendly base field like Fp with p = 2^61-1. But it turns out the discrete logarithm problem (and therefore DH) is strictly easier in these settings (one can exploit Weil restrictions to get an algorithm that is still exponential-time but strictly better than O(p^(g/2)). But this wasn't known until the 2000's.

That leaves g=1 and g=2 as the best choices, and the group law is faster and simpler for g=1, and as far as I know nobody is really working on the g=2 case anymore (but there was a lot of activity in this area 10-20 years ago).