What is this rubber part that dropped out from under my car?

The part you found is almost certainly a critical bushing or damper associated with your front axle assembly, likely a CV axle half shaft. That black rubber piece is meant to absorb vibrations and provide a cushion between the metal components, which is why its failure is causing the violent shaking and banging you described when you hit bumps or turn. Since it is completly torn and missing a chunk of rubber, the mechanical connection is gone and the system is compromised. What you are seeing is a classic symptom of an axle failure. For older Hondas, these rubber components degrade over time, expecially if they've been exposed to road salt or standing water, which accelerates the metal fatigue. When that rubber completely fails, the metal is allowed to clunk against itself, which is exactly what is happening when you turn the wheel or hit anything.

As for wheather you can swap it yourself, yeah, you probly can if you have the right set of wrenches and a jack. It's a common job for anyone who's done basic maintenace before. The whole front axle assembly might need replacing depending on the damage to the splines inside the metal, but the bushing itself is the primary culprit here. You'll need to locate the exact axle for your specific sedan year and engine size to get the right part.

Once you have the replacement, the job involves lifting the car safely and then removing the old, broken half of the assembly. Once the new bushing is mounted, you shold have much better control and the strange noises should vanish. This is a necessery repair to prevent more serious damage down the line.

I agree with the diagnosis that the bushing failure is the main problem. When I had this specific issue on a diffrent ride, the problem wasn't the rubber part itself, but how completely seized the metal components were. Getting the entire assembly out of the housing once the rubber is gone requires a serious amount of torque becasue the metal ussually welds itself to the bracket over time. Be prepared for a tough removal even if the part replacement is easy.

I think you are overestimating how simple it is to swap that piece yourself. While the bushing part itself might seem like a basic cushion you can just replace, the reality of getting the old, broken assembly out of the housing is usually way harder than just using wrenches and a jack. On an older car like this, the metal components bind togehter and rust into each other until they are completely stuck. You usually have to heat the joint or use a proper puller just to get the original piece off the axle. It sounds like a straightforward fix until you actually start trying to remove the seized metal.