Background
At least since Griffiths and Harris [1] we know that the geometric construction "draw the next tangent" appearing in Poncelet's Porism corresponds to addition of a constant in the elliptic curve group. See here for some animations.
The question
A natural follow-up question seems to be that of the question title. More precisely, as highlighted here, because of the dependence of choice of identity, we should probably study the natural ternary operation. Specifically, given a general pair conics $C, D \subset \mathbb{P}^2$, together with points $x_1, x_2, x_3$ in: $$X = \{(p, l) \in C\times D^* ~|~ p \in l\},$$ is there a natural geometric construction corresponding to $x_1 - x_2 + x_3$? I'd even settle for some useful special case, e.g., when $x_2$ corresponds to one of the four points of $C \cap D$.
Further waffle
Thinking about this yesterday I decided that it might be worth looking for natural morphisms from $X$ to $\mathbb{P}^1$ since the divisors of such maps will give identities in the group operation. In a slightly-ridiculous thought experiment, I tried to put myself in the position of somebody with the data of a cubic in $\mathbb{P}^2$ who didn't know the "collinear iff sum to zero" rule for cubics in Weierstrass form but who did know there was a group law, and was searching for a geometric construction. At least in this setting, the idea to look for natural morphisms to $\mathbb{P}^1$ is fruitful since the ratio of two linear forms gives a map to $\mathbb{P}^1$. This gets you most of the way to the "Weierstrass rule", and you just have to spot that you should choose one of the two forms to define a line through an inflection point, which is then a natural identity.
However, try as I might, I couldn't come up with anything useful for $X$ in the Poncelet construction. For a general point $(p, l) \in X$ we have a natural isomorphism $l \simeq \mathbb{P^1}$ since there are three distinguished points on $l$:
- $p$,
- the other intersection point of $l$ and $C$,
- the point of tangency at $D$,
but this seems to be of no use. ¯\_(ツ)_/¯
[1] Griffiths, P., and Harris, J. "On Cayley's explicit solution to Poncelet's Porism", Enseign. Math., 24, 31–40 (1978).
