In the typical Western sense of the word Atheism (that is, not just as a label defining the lack of belief, but the specific instance of ideas/people) is not another "religion," but it's absolutely another "belief system" that often comes with the same set of beliefs about various things.
It's largely an intellectual distinction, because in practice everyone still acts and exists in the world. Identity itself is probably impossible without having some sort of story about who you are.
To be clear, Atheists have different belief systems. They all have in common that they believe in one less god than monotheists, and a handfull less gods than polytheists (that, they share with monotheists), but overall you can have different belief systems that do not involve the existence of one (or multiple) god(s).
I know someone who does not believe in gods because he think aliens are playing with us (fake flags etc. Basically the Stargate mythos, except he never saw the show). Less anecdotal, Chalmers is definitely Atheist, but i sure don't have his belief system, and i'm sure no functionalist does.
Also, in general, philosophy of mind is imho the best way to test you belief systems. Or maybe it's the philosophy field i'm the most comfortable in and thus the one where i pushed my beliefs the furthest :/
I don’t really want to repeat what I already wrote, but just to sum this up again: yes, the mere label of atheist doesn’t imply a specific belief system. But in actual practice, in the real world and not merely in definitions, it tends to indicate certain groupings of beliefs.
In other words, the definitional concept of atheist doesn’t imply a specific belief set (except in the choice of using such a word to define oneself), but the sociological definition definitely does. If we looked at various communities calling themselves atheists, they certainly have beliefs in common.
It's largely an intellectual distinction, because in practice everyone still acts and exists in the world. Identity itself is probably impossible without having some sort of story about who you are.