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", Innov. Incidence Geom. 9 (2009), 79–122 that I wrote together with Yoav Segev, for an introduction to the subject.