discrete geometry - What is the oriented Fano plane?

Here is one answer: It is an oriented line over $mathbb{F}_7$.

An affine line over $mathbb{F}_7$ is a set of 7 points with a simply transitive action of $mathbb{Z}/7mathbb{Z}$, but no distinguished origin. Here, we don't have a distinguished origin and we also don't remember the precise translation action, but we have a distinguished notion of addition by a square (think of what this would mean for real numbers). In other words, it is a set with seven elements, equipped with an unordered triple of simply transitive actions of $mathbb{Z}/7mathbb{Z}$, such that translation by 1 under one of the actions is equivalent to translation by the square classes $2$ and $4$ under the other two actions.

If you take any pair of points $(x,y)$ in the above picture and subtract their indices, the orientation of the arrow between them is $x to y$ if and only if $y-x$ is a square mod 7. Furthermore, a triple of points $(x,y,z)$ with directed arrows $x to y to z$ is collinear if and only if $frac{z-y}{y-x} = 2$. Even though the numerator and denominator are only well-defined up to multiplication by squares, the quotient is a well-defined element of $mathbb{F}_7^times$, since each of the three translation actions yield the same answer. These two data let us reconstruct the diagram from the oriented line structure.

There is a group-theoretic interpretation of this object. The oriented hypergraph you've given has automorphism group of order 21, generated by the permutations $(1234567)$ (one of the translation actions) and $(235)(476)$ (changes translation action by conjugating). This can be identified with the quotient $B^+(mathbb{F}_7)/mathbb{F}_7^times$, where $B^+(mathbb{F}_7)$ is the group of upper triangular matrices with entries in $mathbb{F}_7$ and invertible square determinant, and $mathbb{F}_7^times$ is the subgroup of scalar multiples of the identity. This group is the stabilizer of infinity under the transitive action of the simple group of order 168 on the projective line $mathbb{P}^1(mathbb{F}_7)$. In this sense, we can view the simple group as the automorphism group of an oriented projective line, since it is the subgroup of $PGL_2(mathbb{F}_7)$ whose matrices have square determinant.

Unfortunately, I do not know a natural notion of orientation on an $mathbb{F}_2$-structure. I tried something involving torsors over $mathbb{F}_8^times$ and the Frobenius, but it became a mess.