Why is a constitutional monarchy considered a form of democracy?

The underlying assumption of this question, which is that constitutional monarchies are considered democracies by dint of being constitutional monarchies, is false. "Constitutional Monarchy" simply means that the power of the monarch (king) is limited by a constitution. This constitution can be enforced by some kind of oligarchy, aristocracy, or the common gentry. If the constitution is enforced by an oligarchy or aristocracy, the resulting state is not a democracy, whereas if it is enforced by the common gentry, it is democratic.

Now, since almost all current constitutional monarchies are of this latter variety (monarchies with a constitution enforced by common people), people generally drop the intermediate step of separating non-democratic constitutional monarchies from democratic constitutional monarchies and refer to constitutional monarchies as a form of democracy.