Hear me out. This thought process requires a bit of knowledge of physics/chemistry.
On the martian poles, there are vast quantities of frozes CO2. This frozen CO2 exerts a certain “vapor pressure” - in other words, a certain partial pressure of gaseous CO2.
Now, if we convert this CO2 into O2 by removing the carbon out of it, the concentration of O2 in the atmosphere increases. And therefore, the concentration (and partial pressure) of CO2 decreases.
But since the frozen CO2 on the poles causes a certain partial pressure of CO2, a bit of the frozen CO2 will go into gaseous phase to refill the CO2 partial pressure.
So, by converting CO2 into O2, the concentration of O2 increases, but the concentration of CO2 stays approximately the same. As such, the total pressure (and density) of the atmosphere increases. This would happen if large-scale biological photosynthesis/growth took place.
So, the major issue with settling moon is resource availability: water (!), carbon, fertile soil, and energy.
On the moon you have none of that. Maybe, with a lot of luck, you find water somewhere. Then you need carbon, energy during the long moon nights, and soil that isn’t razor sharp particles.
On Mars, you have all of them: low concentrations of water in the atmosphere, carbon from carbon dioxide in the atmosphere, soil that isn’t razor sharp (thanks to erosion), and the nights are short enough that you can make it through them.
True, and you have more gravity. On the other hand you have perchlorides. Obviously you need water but I thought they did find some already near the south pole on the moon? (EDIT: Yes but apparently little and not very concentrated). I didn’t know there was so little carbon though.