The atmosphere gets the attention because that is where temperature is measured. The ocean has absorbed the large majority of additional heat and roughly a quarter of emitted carbon dioxide.
Without that absorption, surface warming would have been dramatically greater.
Why the capacity may change
Warm water holds less dissolved gas than cold water. As the ocean warms, its ability to absorb carbon dioxide decreases.
Absorption also depends on circulation carrying surface water into the deep. If that circulation slows – and there is evidence of weakening in the Atlantic – carbon-saturated surface water stays at the surface and absorbs less.
Both are feedbacks that reduce the buffer as warming continues.
Acidification
Dissolved carbon dioxide forms carbonic acid, and ocean pH has measurably fallen. That affects organisms building shells from calcium carbonate – which includes plankton at the base of marine food webs.
This is a consequence of emissions independent of temperature, and it receives a fraction of the attention.
Heat in the deep
Heat absorbed by the ocean does not disappear. It is stored, and it comes back out over long periods, which means some warming is committed regardless of future emissions.
That commitment is one reason temperature targets are harder to meet than emissions arithmetic alone suggests.
What is being measured
An array of autonomous floats profiling temperature and salinity throughout the world’s oceans, continuously. That programme has transformed understanding of ocean heat content and is the reason these questions can be asked quantitatively.
Why it matters for projections
Every climate model assumes an ocean uptake rate. If that rate declines faster than assumed, projections based on it are optimistic.