This is a great question! I think it is correct to say "the emissions impact of consumption is roughly proportional to the amount you pay for it", but if you mean "the amount you pay is roughly equal to the environmental impact measured in monetary terms" then the question is a lot more complicated, but I don't think there's a good reason to think that.
An argument for the first claim is:
1) The supply chains for almost all goods and services involve energy produced from fossil fuels
2) Energy is a variable cost, so is roughly proportional to the amount you pay for a good or service
3) Therefore, the emissions impact per dollar spent will generally fall within an order of magnitude across most goods and services
This argument also works (perhaps to a lesser degree) to other intermediate products such as cement, steel and aluminium, which appear in many supply chains and have very high non-energy emissions.
The exceptions to the first claim tend to be products that depend directly on processes that have high non-energy emissions e.g. flying (fuel burning), food (land use from agriculture). This disparity will become even more stark as we decarbonise the power sector (which is easy compared to industrial emissions).
One way to be concrete about this is to estimate the "Scope 3 multipliers" associated with the various products in the economy, which give the emissions impact per dollar/pound/euro through the whole supply chain. One approach to this is to use input-output tables augmented with direct emissions-intensities for each sector (this is what I do a lot in my job), but there are other product-specific things you may want to account for (this seems like a good reference)