One defining feature of $\mathbb P^1(k)$ is that it provides a sharply 3-transitive permutation representation for $\operatorname{PGL}_2(k)$. I believe that the abstraction of projective line to "sharply 3-transitive permutation group" is the most studied one.
The characterization of sharply 3-transitive groups as groups of projectivities over KT-fields came up in an answer to Jacob Lurie's question Action of PGL(2) on Projective Space. That answer mentions that every sharply 3-transitive group is the "group of projectivities" of a KT-field $F$, but note that the correspondence goes both ways, and one can construct $F$ out of the permutation representation.
Judging by how there is an equivalence of categories of near fields and of sharply 2-transitive groups, I wouldn't be surprised if one can say something similar for KT-fields and sharply 3-transitive groups. A reference for this is
"Kerby, W., Wefelscheid, H. "Uber eine scharf 3-fach transitiven Gruppen zugeordnete algebraische Struktur." Abh. Math. Sem. Univ. Hamburg 37 (1972), 225–235.