I think the article's point is that the notion of "groups" per se is too coarse for DH, because it abstracts away the data structure you use to represent the elements of the group. However, the notion of Algebraic Groups doesn't have this weakness, because it comes with an implied data structure: The data structure is tuples over finite fields, and the group operations are expressible as polynomials, which importantly are also algorithms. Done.

It then turns out that if you search for the algebraic groups that you can use for DH, you end up looking for candidates within the 2 possible kinds of algebraic groups: linear algebraic groups and abelian varieties. Within the abelian varieties, it turns out that elliptic curves are both the simplest kind, while also being perfect for DH. Within the linear algebraic groups, the possible cyclic groups are just the additive and multiplicative ones: The additive ones are hopelessly insecure, while the multiplicative ones are just the unit groups of a field, which are special cases of elliptic curves anyway. So elliptic curves really are all you need for DH.

There's a mapping there, however it's not natural.

Why don't some phrases arrive at the same meaning? They don't commute.

You can be wrong a lot and then suddenly be right. Look up "being a Turkey": https://en.wikipedia.org/wiki/Turkey_illusion