Wealthy in the US vs "comfortable" in France?

Preface: I fully realize I'm coming from a very privileged position with this question.

I have the option to leave the US, but am unsure if the change in lifestyle is worth it (or if there would even be that much of a change).

M34 (US/EU citizen) + F32, no kids, 1 dog (20kg), currently making a combined $13,500/mo after tax and health insurance. Apartment in HCOL city.

Looking at moving to a city in France (not Paris), would be making $6,000/mo after tax. Looking at an apartment in the city/downtown. Both of us are bilingual with other languages, so not worried about picking up French.

I know the cost of living in France is much lower than the US, but is it really that much lower? Food and rent may be cheaper, but it seems like things like cars, airfare, etc would still be the same price. Would we be able to have a similar quality of life to our US income? We also live in a very dog friendly city, so all apartments take dogs; is this common in France?

[edit] currently live a pretty laid back life in Colorado, would be moving to Bordeaux.