Does a cooler have to match the CPU or just the socket?

It fits the socket but does it have to support the CPU as well? I want to buy a cooler and i see in the description that other than the socket they also list what CPUs are supported.

You should check your other post. It is best to only post a question in one place.
That being said;

Some coolers make reference to the CPU as well as the socket since the heat sink must be able to dissipate the heat created by the CPU. While different CPU's may use the same socket, one may need a better cooler to keep the temperature under control if it uses more power.

The cooler should be one made for your CPU.