It is right because all the planet cares about is the amount in the atmosphere.

It is completely wrong because whatever total amount X of CO2 emissions we decide we need to limit humanity to, we have to somehow allocate that CO2 emission budget to people.

Unless you can make a good case that some people have some sort of natural or divine right to a bigger share than others the only way to do it is for every person to have the same share. (Maybe on top of that there can be some system of trading, so that people whose emissions go toward things that the world finds more useful can trade for a bigger share, but it would still have to start from everyone is equal).

Since it is countries that would be the ones actually enforcing whatever the limit is, the country effectively ends up as the entity in charge of the shares of all its population, making the allocation at the country level equal on a per capita basis between countries.

Another way to see that anything other than per capita is nonsense is that anything else would allow groups of people to manipulate their allocation by splitting the country into multiple countries or merging multiple countries into one.