To expand on your other idea, for the case of a metric space we would want "collinearity" to resemble an equivalence relation. That is, three points $x,y,z$ will be called collinear if either (1) at least two of the points are actually the same point, or (2) if we rename the points so that the largest of the three distances is $d(x,z),$ then we require $d(x,z) = d(x,y) + d(y,z).$ The part about equivalence is just this: we require that if $(x,y,z)$ are collinear, and then $(x,y,t)$ are collinear, we require that the other two triples are also collinear: $$ (x,z,t), \; (y,z,t). $$
It is always possible that this kind of metric space has a name. If finite, it should be possible to isometrically imbed this in a Euclidean space. But I am not sure what happens if you define such an object and allow it to have infinitely many points. Hard to say, I just made it up.
Anyway, a metric space with Mariano's discrete metric cannot have this property. That is, there is no acceptable way to define "collinear" in that case.
P.S. Mariano and Fernando Q. Gouvea are the same person.
P.P.S. As pointed out by Mariano Suárez-Gouvea-Alvarez, even in the case of the 7 point plane with each uninterrupted segment having length 1, the result cannot be imbedded isometrically in a Euclidean space. Live and learn. The reprint by Wojtech Skaba does not include a metric space structure consistent with his collinearity structure, so this is slightly new territory, maybe there is something amusing there.