Why I might get this high noise on Sony a7 III with Zeiss 24/70 f4 lens?
You need to define what is acceptable to you.

Record a well illuminated scene and increase the ISO while decreasing the aperture and compare what noise level on your specific camera is acceptable for you, and you will know what are the limitations of your equipment.

In the case of the example video, the noise is "the same" because both moments, the dark zone and the white are filmed in the same settings, therefore we can assume that is the overall noise on your camera on that ISO setting.

There are some programs than can denoise video https://duckduckgo.com/?q=video+denoiser&amp;t=h_&amp;ia=web one that has a good reputation is Neat Video.