Obviously not in name, but in theory.
The UK is a representative democracy; its administration is elected by the people, who are recognised as citizens, not subjects, enjoying the various freedoms (speech and so on) that that implies, without an obligation of fealty. The unelected house cannot prevent the elected house from carrying out its will if it's determined enough, and the unelected monarch has limited executive powers which can be nullified or taken away by Parliament, and effectively go unused. Parliament has also demonstrated a low tolerance for dissent from the monarch in the past.
In other words all of the same basic freedoms and mechanisms seem to be present that would be in a parliamentary republic. So from a theoretical point of view, should the UK (and other "ceremonial monarchies", e.g. Japan) be classified with, and regarded as, republics (rather than linking them with true executive monarchies)? Are there any concrete implications to this?