This linear assumption is done all over the place in science and engineering. Sometimes, it works. Not always.
For example, consider a spring. It has a force
F = k*x
...but does it? No. Actually it has a force:
F = k_1*x + k_2*x^2 + k_3*x^3 + ...
x is small enough, the linear assumption is good.
Similar assumption is done in optics. It is there assumed that the material responds linearly to the electric field of electromagnetic waves. Usually it does. However, if the electric field is very strong, like in very intense lasers, it's possible to get nonlinear behavior in optics.
So, as long as market movements are small, linear assumption works. It's a model. It has a validity limit. When blood starts flowing in the streets, the validity breaks down. When Fed starts printing money like in Zimbabwe, the validity also breaks down.
This linear assumption is justified by the fact that every analytic function can be calculated by an infinite Taylor series. Assuming
x is small, a good enough approximation is throwing all but the first term away, so in that case, the function is simply:
y = y_0 + k*(x - x_0)
...besides, I think this question might actually belong to Quantitative Finance instead of here.