When I perform parametric modeling, if there is significant multicollinearity between variables I think should be independent, but in fact are not, I run into the case where one or more of the coefficients becomes exceeding small (or large) relative to the others. How is that different than what occurs in fine tuning problems of the standard model?
The difference is, that if a quantity in your model is secretly theoretically zero, in practice, you will typically still get a small non-zero number, whose size is determined by the scale of uncertainties in measurements, rounding errors, etc.
On the other hand, in a naturalness problem in fundamental physics, we typically have a dimensionless quantity, whose measured value is significantly bigger than the error-bars (and therefore a non-zero quantity!) but much much smaller than $1$.