I agree that ranking the weights from 1 to N is idiotic because it doesn't respect the relative importance of each characteristic. However, changing the ratings from 101-110 for every scale will just add a constant to each option's value:

• Option A, strength 103, mass 106, total score 2(103) + 106 = 312
• Option B, strength 105, mass 103, total score 2(105) + 103 = 313

(I changed 'weight to 'mass' to avoid confusion with the other meaning of 'weight')

Using something approximating a real-valued ranking (rank from 1-10) instead of rank indicies reduces the problem to mere nonlinearity.

I assume you mean using values for the weights that correspond to importance, which isn't necessarily 1-10. For instance, if strength is 100 times more important than mass, we'd need to have weights of 100 and 1.

You're right that this assumes that the final quality is a linear function of the component attributes: we could have a situation where strength becomes less important when mass passes a certain threshold, for instance. But using a linear approximation is often a good first step at the very least.