Why does it do this?

98 Mitsubishi Eclipse, 2.0L non-turbo, 5-speed. Background: The car is parked on a slight incline facing downhill. Every morning when I start it, it idles at 1000 rpm standing still. It’s in neutral and I let off the parking brake and let it coast very slowly (1/2 mph) and it stays at 1000 rpm. If I let it coast up to 3 mph (still in neutral) it will suddenly speed up to 2000 rpm and stay there until I put it in gear to start driving off. Why is this? Is there something in the computer that detects the speed and speeds the engine up?

" Is there something in the computer that detects the speed and speeds the engine up?"
Yes. (I assume you have a manual transmission.)

yes, it’s a 5-speed…and it does this without pushing in on the clutch or putting it in gear. I can creep down the driveway and it will stay at 1000. If I let it speed up past a couple mph, it goes to 2000 rpm.

There is a speed sensor on the output shaft of your transmission. You computer is aware the car is moving and it must get a little confused…

I doubt that it is confused. It is getting ready for the OP to put the car in gear.