I checked the tag. Five of six questions used DNC correctly to refer to the committee organization. I edited the tag out of the sixth (and democratic-party would not have made sense there). It was a Barack Obama administration question, not a party question.
Anyway, those five questions are actually about the party committee rather than the party itself. The reason why we have dnc and not rnc may simply be that the RNC has been less interesting.
The tag does not seem to be causing much confusion (as I said, the one misuse would have been just as misused if the synonym were there). And the DNC is an organization, separate from the party. The democratic-party tag is often used as a surrogate for Democrats. The DNC is not.
Part of the confusion is that democratic-party is very seldom used to refer to the party itself. Instead, it is used to refer to the Democrats as a (for lack of a better word) demographic. I.e. that group of people who consider themselves Democrats. Occasionally it is used to refer to Democratic politicians.
The DNC does not set the Democratic agenda. Currently that's Nancy Pelosi and Chuck Schumer to the extent that it is set. And some things are being set by individuals, e.g. Bernie Sanders' Medicare for All plan and Alexandria Ocasio-Cortez's Green New Deal.
It would be more accurate to describe the DNC as the fundraising and event planning part of the Democratic party. Except that contribution limits mean that they actually try to make as many organizations as possible so as to evade the limits. The DNC is the most official national organization for fundraising and event planning not associated with a specific chamber of Congress (as are the DCCC and DSCC).
It has six questions (I added another while investigating) that are legitimately about the DNC and not so much about Democrats more generally.