Mars is actually highly oxygenated already - that's why it's red. Also all the salts on the surface are highly oxygenated as sulfates, chlorates and perchlorates instead of the chlorides and sulfides we get on Earth -those hardly occur at all on Mars. While the other way, chlorates and perchlorates are rare in salt deposits on the Earth. So you could say, Mars is more oxygenated than Earth!
Took many hundreds of millions of years for the Earth to oxygenate all its iron deposits in the Great Oxygenation Event. Mars does have a small amount of oxygen in its atmosphere at just over the 1% level. It may have had an oxygen rich atmosphere early on, if so that could have been caused by photodissociation of water into hydrogen and oxygen with the hydrogen lost to space. Mars Had Oxygen-Rich Atmosphere 4 Billion Years Ago, Shows New Study
The thing is that oxygen is a very reactive gas, easily reduced if you have any reducing agents.
So, given that we also have many reducing gases, then in the absence of photosynthetic life to oxygenate our atmosphere, we would lose all our oxygen in 4500 years.
In case of Mars, then the main difference from Earth as far as composition is the lack of water.
Probably Mars never had quite as much water as Earth. It's now thought that it had an ocean over the entire Northern hemisphere - but not as deep and extensive as the Earth's oceans Water on Mars and Life
Is a big mystery, what happened to that water. Was it lost to space, or is it still there, perhaps as ice in the Northern hemisphere below the planes? Is evidence for at least some ice below the surface there. But hard to see how all that water could vanish below the surface, so perhaps some lost to space also. The MAVEN mission may help by doing detailed measurements of loss of molecules from the upper atmosphere.
In any case the equatorial regions are thought to be dry to considerable depths, for theoretical reasons, to kms below the surface.
Probably is a subsurface layer of water, could be a few hundred meters thick or more, called the "hydrosphere" several kilometers below the surface, heated by geothermal heating, and kept under pressure and trapped by the overlying rocks.
If we were to add water to Mars in some way, as in some terraforming proposals, then most of the water to start with would sink below the surface in the equatorial regions - it would be like pouring water into the Sahara desert but more so - the Sahara has subsurface aquifers, but Mars probably doesn't, except possibly localized "oases" of trapped ice which have been detected in equatorial regions. But most is thought to be completely dry for depths of kilometers.
On the other hand the Southern uplands are at a higher altitude, so - perhaps if they had lots of ice trapped in the higher latitudes, that might compensate for the effect of water getting absorbed in equatorial regions. Don't know if anyone has done the calculation, or if we have enough data to be sure either way.