There's safety, exhaust emissions, tyre and brake particles, noise, etc. I'm sure some of them don't have a linear relationship either.
One of the biggest upsides to taxing vehicle weight could be to counter purchasing decisions that have shifted to larger vehicles that no longer comfortably fit in parking spaces but ride better over ridiculous speed bumps and give drivers literally the opportunity to look down on others.
Taxing vehicle weight also doesn't have the privacy implications that taxing miles driven beyond fuel consumption does. It's always struck me as one of the more sensible aspects to tax.
Add lightness, as I believe Colin Chapman put it.