Why isn't the word Atheist a proper noun when Christian and Islam are?
I think this is more subtle than it appears:
"Islam" is a proper noun because it is the name of a unique thing - the Islamic faith.
"Christianity" is a proper noun for similar reasons.
"Christ" is considered a proper noun as it is treated as the name of one particular individual. As a word it really means "anointed one" (from the Greek language) and corresponds to the word "messiah" that we get from Hebrew. As such it is more of a description than a name. I think we treat it as a name by tradition.
"Christian" was a term coined at Antioch, meaning "little Christ" or "little anointed one" - meant to be a mildly disparaging term, but accepted and adopted by those who followed Jesus. I am not sure that it is really a proper noun. I think we capitalise the first letter due to the reference to Christ.
"Atheist" is a general term (also from Greek) basically meaning someone without any god. It describes a characteristic of an individual but does not name an individual. So it is not a proper noun.
On the other hand "Atheism" could be considered a proper noun in that it could be considered a faith in the non-existence of God or gods. However, it can also be considered to simply express lack of faith, which would make it not a proper noun.
Because it isn't a proper noun.
Religions are proper nouns. Proper nouns are defined as an individual person, place, or organization and because religions are names of organised groups, when you say that someone is a Christian or a Muslim (or any other religion for that matter) you're ascribing them, or their belief, to a specific organization. I feel it's also worth noting that proper nouns are always capitalized, unlike common nouns, which are only capitalized in certain circumstances. Now in order to understand why we don't consider the word atheist to be a proper noun we have to break the word down into its constituent parts.
In English we like to steal words, including prefixes from other languages (we're a rapscallious horde of grammar stealing hood rats).
One word where this grammar stealing concept holds true is the word atheist, where we stole the Greek prefix a- meaning "not".
Now someone that's a theist believes in a God who "made and governs creation".
So if we place the prefix a- in front of theist we get the term atheist. An atheist is someone who doesn't believe in a God, essentially they lack a belief in the existence of a God or gods (they don't believe in anything pertaining to the godly).
Since atheism is not an organized group (there might be gatherings of atheists but they don't have an organized doctrine) but rather a belief it doesn't qualify as a proper noun and isn't capitalized. This same capitalization rule translates over to the person who identifies as an atheist simply because atheism isn't an organization or a even really a religion.
I hope this helps!