"Democratic" is the correct adjective, not "Democrat." It would be like me calling somebody a member of the "Republic party."
Actually, it is the Democrat and Republican parties. Of course, the Democrats love being called the "Democratic" party. It conjures up all of those wonderful feelings of democracy in action, the democratic majority, the democratic process, etc., etc., etc. Yet, if you have a preference, you declare yourself as a Democrat or a Republican. Not a "Democratic" or "Republicratic". But, as all good Democrats believe, it's the words that matter much more than the subject or the action. The label is what matters, not what the label is put upon or describes. And thus, we have political correctness in action. Yet, it is great to live in a democracy where all can air the opinion they wish without fear of reprisal or retribution. Oh, but wait, we happen to live in a constitutional republic. Perhaps your logic is correct, and a name change is relevant in these circumstances. So then we could have the Democratic Party and the Constitutional Republic Party. Wonder how the Democrats would respond to that?