Ideas about who might legitimately represent "the people" in a "rule of the people" hadn't moved on much from the Athenian model in, say, England by the 19th century.
You still had to be male and own your own land if your voice was to count for anything.
An explanation of how and why Cardassia ended up as a stratocracy is given by the character of Gul Madred in the TNG double episode Chain of Command
. According to what he tells Picard, the Cardassian people turned to their military for strong leadership and security in times of hardship and trouble, then continued to support the state's expansionist policies because this was the one way they could see to secure the necessary resources and raw materials Cardassia required.
It's hinted in a couple of DS9 episodes as well as in a couple of tie-in books that the civilian Detapa Council used to have more autonomy, but had this gradually eroded until the civilian government representatives were reduced to more or less rubber-stamping legislature insisted upon by the military leadership.