Does using an Aeropress keep more of the coffee oil (crema)? I've noticed that coffee made with my Aeropress has a much richer flavor even though I use the same water temperature, number of scoops, roast, etc.

Is this because the Aeropress method produces crema (the tawny substance found in espresso shots)? And, if so, is there a way to optimize crema extraction with an Aeropress?
The aeropress does not produce crema. There is no where near enough pressure using the aeropress to produce crema. Crema doesn't make coffee better, its typically a sign of good espresso, but isn't found in any other form of brewing nor is it what make the coffee actually taste better. French presses don't create crema either, and its a misleading yet common belief that they do.

The aeropress likely has about the same amount of oils as a pour-over cup of coffee if you're using the paper filters. The aeropress get it's richness from its style of brewing known as "immersion brewing". Longer contact with the grounds extracts compounds usually left behind by pour-over/drip methods. Combine with pressure, the aeropress, in short, just extracts more flavor by having a longer brew time and an environment which increases solubility with pressure.

Clarification:

The AeroPress has a short "brew time", however what I was meaning to say is that the aeropress and other immersion brewers have a longer contact time of the bean and the water. Therefore you have more extraction. The aeropress, unlike any other immersion method uses pressure to increase solubility. It's known that if you increase pressure that you extract quicker, and your solvent(water in this case) is able to achieve higher saturation.