I think the idea is that you're powering traditional combustion engines with a fuel generated with green power. It may be more wasteful of that green electricity than a new EV charging from the grid would be but you didn't have to build and buy a new thing and it's kind of fine to waste renewable energy if the economics works out because it's renewable.

So it's effectively a way to convert your ICE car into a zero-emission EV by using that fuel.

The energy content of the gasoline is less than the energy used to make it (only 50-60% efficiency), but the CO2 released is the same as the CO2 used. So the energy efficiency is low, but the net CO2 emission is still zero.

Is your device small enough that it could be placed to filter the exhaust emissions from a fuel burning car and reinsert the newly created gasoline into its fuel tank?

powered by what? It would only make sense if there was a carbon free cheap power source you could use to charge it - if you just take it from the car battery you're wasting fuel as it's only ~50% efficient