While not completely explained in the article, the author correctly notes that the non-singular points on a singular cubic curve over a finite field form a group (under the same elliptic curve group law) that is isomorphic (in an easily computable way) to either the multiplicative or additive group of a finite field.

You can find more details in Silverman's book on elliptic curves, or if you don't have access to that, see Section 24.7 in Lecture 24 of https://ocw.mit.edu/courses/18-783-elliptic-curves-spring-20....

One could nitpick by pointing out that singular cubics are not elliptic curves (which are, by definition, smooth projective curves). One could also nitpick that the real reason you don't want to use DH in the multiplicative group of a finite field is that there is a known subexponential time algorithm that breaks DH in that setting (forcing > 3000-bit key sizes) whereas for elliptic curves the only known attacks take exponential time (and 256-bit keys are fine). But I don't think either of these nitpicks contradicts the author's thesis.