I shook my camera and lens very hard... Can that cause damage?

Well does your camera work now? I don't think there should be any damage, unless you drop it on the floor or something like that

Cameras are not that easy to break by "shaking"

Your fear that it might not be the best way to handle a camera, is quite right. Although most cameras and lenses are astoundingly robust and can be used in rough situation without much danger of damage, there are limits. Mild shaking, however should not be any problem.

Shaking something violently is similar to dropping on the floor as there will be quit abrupt changes in the velocity, which applies a lot of energy that has to go somewhere. You know that from doing an emergency halt in a car.

This can be a problem for moving parts in your camera. When things are in motion, these things want to keep that motion (conservation of momentum). The urge to keep the motion multiplies with its mass. Suddenly changing the direction (shaking) will apply forces, that can potentially break things.

So lightweight things like the shutter will have less energy applied than heavier things. Luckily most things in your camera are quite lightweight - apart from the in-camara stabilizer (IBIS) if your cam has one.

This CAN be damaged by shaking, especially when it is not locked. This is usually the case when the camera is switched off. The IBIS then can move around and when you apply more force than the centering can handle, it will crash into the side of the housing.

With lenses, the situation is similar. It depends on the construction of the lens. Some have floating elements in the tube, which might get damaged. Some might have stabilizers that could be damaged. And the risk of damage greatly changes with the direction. Usually directions 90 degrees from the axis the light passes through is less problematic. Shakes along this axis might be more dangerous.

TL;DR: Stop shaking your gear. It is delicate. You might damage it, if you do it violently enough.