Which stage is best for getting maximum coins?

I always use the Motocross Bike on the moon stage.

When fully upgraded the Motocross Bike spins very quickly (courtesy of the Mid-Air Control upgrade), and when combined with the low gravity of the moon you can get up to 20,000 coins each time you leave the ground.

The moon stage is also described as good for farming (coins) on the wiki:

Its main characteristic is low gravity. It is also very useful for farming as air times are usually 10000+.

Image from the wiki

I haven't played in quite a while, but I always found the race car on the highway stage to be a pretty fast way to gather a lot of coins. Quickly reaching the high-value coins seemed to be a pretty efficient way of coin farming.

You may also want to look at the answers on this previous question regarding the best vehicle on each stage, since distance directly corresponds to coins earned. (though it doesn't directly answer coins earned over time, which seems to be more along the lines of what you're looking for).

Moonlander on Moon Level.

Right as you start the level just thrust straight up until you reach about 1/4 on Fuel Tank. Just release and fall and you'll end up with 35,000 to 50,000 points within about 20 seconds. Rinse, repeat. You'll run about 1 million points every 8-9 minutes.