Oy. Not this again. Look. Theism consists of dogma, rules for behavior, and often enough, a strong and well solidified political agenda, for instance, as with Islam or the Christians that are constantly attempting to fiddle with the sayings on money, messing with the pledge of allegiance, praying in congress before making laws, seeing to it the rest of us can't buy beer on Sunday, etc. They do these things because they think this is the way to "bring" their religion, and its dogma and rules, to the rest of us. Speaking generally, theism is a belief in a god or gods, and it carries, in a very official and intentional manner, a great deal of imposed behavior and canned rules with it.
Definition of ATHEISM
archaic : ungodliness, wickedness
a : a disbelief in the existence of deity b : the doctrine that there is no deity
Please don't add more to the definition than what it is. Atheism is a religious concept. The separation of church and state does away with religious interference in secular laws. The separation of all pressure and interest groups still does not occur even in secular societies because things such as financial resources can influence political decisions.
The Soviet state was ATHEIST according to the above definition. It had an ideology which was not followed properly as the Soviet State was not even truly Communist, it resembled a fascist state. In terms of ideologies, Sweden a secular state, can also be considered to be an ideologically run state where it is considered that democracy, socialism and a myriad of feminist laws (among others) dominate government ideology.
As for voting along secular lines, most religious people are more than capable of doing that. Perhaps some people cannot grasp this but it happens and it works. People even in atheist societies would vote according to subjective opinion. For example poorer, less educated people would want more social welfare while the rich would want more tax breaks and those with children would want a larger budget for education and so on. In every society there is a difference of opinion and people are egoistical and will vote according to what is better for than and not think of the good of the state. Remember the Soviet Union was conceptualized this way - it was meant to be an utopia of altruism, people were repeatedly told that Communism was just round the corner but it never came. True Communism would mean that all form of government would be abolished and a workers'/peasants' paradise would follow with each worker donating according to his abilities and receiving according to his needs. This never worked out, partly because without incentive people will give as little as possible and take as much as they can.