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.

The groups $U_x$ are called the *root groups* of the Moufang set, and the group $G$ is called the *little projective group*.

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 over fields 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).

At 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.

**Edit**: As requested in the comments below, I am adding some more details, in particular about the example with $X = \mathbb{P}^1(k)$ and $G = \mathrm{PSL}_2(k)$.

First of all, notice that the little projective group $G$ is generated by any two of the root groups, $G = \langle U_x, U_y \rangle$ for all $x,y \in X$ with $x \neq y$. So to give an explicit description of an example, it suffices to describe two of these root groups; all others are then obtained by conjugation inside $G$.

We now take $X = \mathbb{P}^1(k) = k \cup \{ \infty \}$, acted upon by $G=PSL_2(k)$, the elements of which I will denote with matrices with square brackets (determined up to a non-zero scalar), so
$$ \begin{bmatrix} a & b \\ c & d \end{bmatrix} .x = \frac{ax+b}{cx+d} \quad \text{for all } x \in X.$$
Notice that $\operatorname{Stab}_G(\infty) = \left\{ \begin{bmatrix} a & b \\ 0 & a^{-1} \end{bmatrix}\right\}$
and $\operatorname{Stab}_G(0) = \left\{ \begin{bmatrix} a & 0 \\ c & a^{-1} \end{bmatrix}\right\}$.
We now define the root groups $U_\infty$ and $U_0$ to be the group of unipotent elements of these point stabilizers, i.e.
$$ U_\infty =\left\{ \begin{bmatrix} 1 & b \\ 0 & 1 \end{bmatrix} \mid b \in k \right\},
\quad U_\infty =\left\{ \begin{bmatrix} 1 & 0 \\ c & 1 \end{bmatrix} \mid c \in k \right\} .$$
The point is that it is now possible to forget about the matrix representation and even about the original ambient group $G$ all together, and only retain the corresponding permutations of $X$.
We then get
$$ U_\infty = \{ x \mapsto x + b \mid b \in k \}, \quad U_0 = \{ x \mapsto (x^{-1}+c)^{-1} \mid c \in k \} . $$
It is now not so hard to imagine that this description makes sense for more general algebraic structures than commutative fields only. And indeed, this works equally well for skew fields, octonion division algebras, and even more generally for Jordan division algebras.
To make examples with non-abelian root groups, similar ideas make sense by replacing the multiplicative inverse by more complicated maps that "behave like a multiplicative inverse".

Another relevant comment, related to your first "ideal property": in the case of skew fields, for instance, it is only possible to recover the skew field *up to opposition*, i.e., in the case of $\mathrm{PSL}_2(D)$, we can recover the pair $(D, D^{\mathrm{op}})$ from the Moufang set, but not $D$ itself. (Here, $D^{\mathrm{op}}$ is the skew field with same underlying additive group as $D$, and with multiplication given by $x*y := yx$.)