Illuminance is "the amount of luminous flux per unit area."

Suppose the 100mm lens was aimed at a large wall, and suppose distance was so that the lens sees a 10 x 10 foot area of that wall, reflecting illuminance back to the camera. OK, so this is a square, but it is about the subject, not the sensor (area matters, shape does not).

Then the 200mm lens would see a 5 x 5 foot area of same wall, half as wide and 1/4 as much area, and so 1/4 as much illuminance.

However, then the 2x diameter (4x area) of f/4 aperture of 200 mm will let 4x the light though at same f/4, therefore f/4 is f/4 exposure, regardless of focal length. 4 x 1/4 = 1 (same).

This is why we use the system called f/stops with the funny numbers, so f/4 will be f/4 and have meaning to us.

FWIW, not asked yet, but about the same argument is the reason that camera distance from the subject does not affect the exposure. The mountain is the same Sunny 16 daylight exposure regardless if we are on it, or 25 miles away. The Sun is special (at 93 million miles), but this also applies to subjects not illuminated by the Sun).

When the subject is seen by the camera at greater distance, that lighted object area also appears smaller. When ten times more distant, the subject dimensions are only 1/10 size, which is 1/10 x 1/10 = 1/100 the area. Inverse square law says the light is 1/100 as bright at 10 times distance. So 1/100 the light in 1/100 the area is the same apparent intensity, per unit of area. It exactly balances out, same exposure. Distance from camera does not affect exposure. Distance from a flash does matter.