The Atari 2600 had a digital joystick. The Atari 5200 was completely analog. If you tear the controller down, there's a complex mechanism that links the stick to two rotational potentiometers. It's similar in outcome to modern analog thumbsticks, but the mechanism is completely different (and is MUCH bigger).
The 5200 joystick didn't self-center, though. The only centering force was from the rubber boot. This ended up being advantageous in a game like missile command, but is widely seen as the Achilles heel of the controller.