I think I know that the atmosphere contains a constant oxygen content of 21%, what mechanism is behind this despite constantly changing circumstances?

Answer

The concentrations of atmospheric gases, of course, are always changing.
You compare the oxygen content to the carbon dioxide content. You say carbon dioxide has been increasing lately. The current level of carbon dioxide is about 380 ppmv, in other words 380 ‘parts per million parts’ (ie about 0.04%). The oxygen content, on the other hand, is 21% or about 210,000 parts per million parts. You see that an increase in the carbon dioxide content does not really have much influence on the global oxygen content.
How does oxygen get into the atmosphere? Mainly through photosynthesis. But it’s not just trees and other land plants that photosynthesize and produce oxygen; many unicellular organisms also take part in photosynthesis; they are mainly the oxygen producers. We are thinking here, for example, of cyanobacteria; it is these organisms that more than 2,000 million ago “invented” photosynthesis and released oxygen into the atmosphere.