Following up on Matthias Wendt's comment, the language of Moufang sets is indeed a suitable axiomatic approach to (generalizations of) projective lines.

Formally speaking, a Moufang set is a set $X$ together with a collection of groups $U_x \leq \operatorname{Sym}(X)$ (one group for each $x \in X$), such that:

 - each $U_x$ fixes $x$ and acts sharply transitively on $X \setminus \{ x \}$;
 - the group $G := \langle U_y \mid y \in X \rangle$ permutes the $U_x$'s by conjugation.

It is not hard to see that $X = \mathbb{P}^1(k)$ and $G = \mathrm{PSL}_2(k)$ has the structure of a Moufang set, but there are many more interesting examples, most notably those arising from semisimple linear algebraic groups of $k$-rank one.

It is possible to single out the genuine projective lines among the Moufang sets. For fiels $k$ with $\operatorname{char}(k) \neq 2$, this was done by Richard Weiss and me (see section 6 of our paper "Moufang sets and Jordan division algebras",
*Math. Ann.* **335** (2006), no. 2, 415–433);
this result has been generalized to arbitrary fields (including the case $\operatorname{char}(k) = 2$) by Matthias Grüninger, where the result is somewhat more subtle ("Special Moufang sets with abelian Hua subgroup",
*J. Algebra* **323** (2010), no. 6, 1797–1801).

With the risk of giving too much advertisement for my own papers, I can recommend the course notes "[A course on Moufang sets][1]", *Innov. Incidence Geom.* **9** (2009), 79–122 that I wrote together with Yoav Segev, for an introduction to the subject.


  [1]: http://iig.ugent.be/online/9/volume-9-article-2-online.pdf