There are different ways to think about it, and I don’t quite get what you mean about 2 separate PD controllers, but that might also be a valid analogy. This is partly why I qualified my comments about the I and D terms with “sort-of” ;). But I’m also trying to boil it down to the simplest analogy possible, and I might sacrifice a little precision along the way. The mechanism and forces of PIDs do not exactly correspond to physical suspensions, but if you stand back and look at it, they are surprisingly similar at a conceptual level. The most important part is that the problems are the same, it’s a system that’s trying to smooth out the bumps and prevent overshoot. PID controllers are more abstract than suspensions, and they can be used in ways that feel different, but the basic problem is more or less one of absorbing shocks smoothly. Like your oven or your thermostat, they’re trying to keep the temperature you set and minimize overshoot and oscillation, even though they only have a binary on/off output mechanism.
Here’s how I think about it: the P term is the spring. In a physical spring the force is quadratic in position or “error”, while a PID controller’s P term is linear. No reason you can’t make a P controller quadratic or anything you want, but the analogy between P and the spring seems clear enough to me.
Now compression damping is trying to slow down or resist fast compression and prevent overshooting, prevent having too much spring force from accumulating on the rebound. Sometimes there are even separate high and low speed compression damping mechanisms (I’m sure you know that). In a PID controller, the D term is also designed to resist proportional changes from happening too fast, it resists change in proportion to how fast things are changing.
And back to physical suspension again, rebound damping is designed to prevent rebound force from accumulating. You want to return to neutral position as fast as possible without a rebound force. The idea is to balance 2 different goals: 1- don’t buck you off the bike, and 2- don’t let the suspension pack down and stop working. The PID controller’s I term does this, in a way, it prevents the pack-down by monitoring the average position and if it stays too far away from neutral for too long, it pushes things back to neutral.
If you’re paying attention, you might notice I swapped I & D in my analogy here compared to above. It can (sort-of) be argued either way, which maybe goes to your point that the analogy ain’t perfect, and you could maybe look at it as 2 PD terms. I think it’s best not to try to fix the analogy, but just to see it as a very blunt instrument for understanding - a suspension and a thermostat and a robot steering a car all have a common type of problem to understand and at least conceptually, a solution that looks similar from a distance.