Why does the metric have to be bilinear? Given a spacetime displacement $\textbf{x}$, we can define the interval $I(\textbf{x})$ as the square of the time measured by a clock that moves inertially along $\textbf{x}$. If we assume that the interval can be derived from a bilinear function $f$ as $I(\textbf{x})=f(\textbf{x},\textbf{x})$, then the ability to measure $I$ implies the ability to measure $f$ as well. That is, if you have a norm, and you assume it comes from a bilinear inner product, you automatically get an inner product for free. Or in more physical terms, if you have a clock and a way to tell whether a world-line is inertial, you have a way to measure the metric.
But is there some nice physical or mathematical way to see that the interval should be derivable from a bilinear function? If we rule out the degenerate Galilean case, then $I$ must be compatible with Euclidean geometry for spacelike displacements, and the Euclidean metric is bilinear. So this makes it kind of plausible that the spacetime metric should be bilinear as well. But is there any really nice way to show that it has to be bilinear?
The bilinear form of the Euclidean metric is basically the Pythagorean theorem, which is a statement about parallelism. Is the bilinearity of the spacetime metric interpretable in some nice way as a statement about parallelism?
 A: This isn't anywhere near an answer... but just a possible guide into the literature.
On my list of things to read (now pushed further back) are papers on the foundations of spacetime geometry along the lines of the Ehlers-Pirani-Schild (EPS) approach, which tries to motivate the Lorentzian structure of spacetime.


*

*Republication of: The geometry of free fall and light propagation
Jürgen Ehlers, Felix A. E. Pirani, Alfred Schild
General Relativity and Gravitation
June 2012, Volume 44, Issue 6, pp 1587–1609
https://link.springer.com/article/10.1007/s10714-012-1353-4

*Editorial note to:
J. Ehlers, F. A. E. Pirani and A. Schild,
The geometry of free fall and light propagation
Andrzej Trautman
https://link.springer.com/content/pdf/10.1007%2Fs10714-012-1352-5.pdf
Along those lines are approaches to consider a Finslerian geometry


*

*A spacetime primer
T. A. Jacobson
http://terpconnect.umd.edu/~jacobson/spacetimeprimer.pdf


One such
  attempt appears in a classic paper by Ehlers, Pirani and Schild (EPS), which
  develops a system of axioms for spacetime structure in terms of topological and
  differential axioms about the properties of freely falling massive and massless
  point particles.
  One deep question is why the causal cone is given by a quadric in the tangent
  space...
  
  ...From time to time people try to generalize the notion of the spacetime
  metric to allow for non-quadratic line elements. These go under the rubric
  “Finsler metrics”.


*Finsler Geometry Is Just Riemannian Geometry without the Quadratic Restriction
Shiing-Shen Chern
Not. Amer. Math. Soc. 43, 959-963, 1996.
http://www.ams.org/notices/199609/chern.pdf

*Finsler spacetime geometry in Physics
Christian Pfeifer
International Journal of Geometric Methods in Modern Physics, Online Ready, 2019
https://arxiv.org/abs/1903.10185
https://www.worldscientific.com/doi/abs/10.1142/S0219887819410044

*Light cones in Finsler spacetime
E. Minguzzi
Communications in Mathematical Physics
March 2015, Volume 334, Issue 3, pp 1529–1551
https://arxiv.org/abs/1403.7060
https://link.springer.com/article/10.1007%2Fs00220-014-2215-6
Some links to the literature:
http://www.phy.olemiss.edu/~luca/Topics/geom/finsler.html
Happy hunting!
