The conventions on the capitalisation of wild cat names has altered slightly over the years. Currently, ‘mountain lion’ should not be capitalised because both words are everyday words which are not capitalised in normal use. The names of wild cat species are only capitalised when they contain place names such as the Bengal tiger, Asiatic golden cat or Chinese mountain cat.
In fact, none of the various names of the mountain lion are capitalised. Some of the more common names of this most named of all wild cats are puma (the real name), cougar, panther and catamount – all lowercase.
In 1996 Dr Desmond Morris did capitalise all the names of the wild cat species. So, for example, he capitalised ‘Puma’ but today we write ‘puma’. He capitalised ‘Lion’ whereas today we write ‘lion’ and so on. Dr Morris is a world renowned English zoologist and author. What he did was not incorrect. It was his choice and at that time the convention was more towards capitalisation.
On this page, directly below are two photographs showing the capitalisation and non-capitalisation of ‘puma’ and ‘mountain lion’. The first is from Dr Desmond Morris’s book Cat World (published in 1996) and the second (published in 2002) is from Fiona and Mel Sunquists’ book Wild Cats Of The World (the best book on wild cat species).
In my view there is some flexibility on whether you capitalise ‘mountain lion’ or not. The modern convention, as mentioned, is not to capitalise the name but if you do, no one is going to criticise you. The English language is not cast in concrete. It is constantly changing or evolving and as with grammar you can apply discretion sometimes. You can even make up new words and they’ll be acceptable.