I think that the Democratic Party is hated both from the left and the right, and calling them the Democrat Party makes some sense if you don't view them as aligning with real democracy. Kind of like when one calls pro-life people "pro-forced-pregnancy" or "anti-abortion".
Like there's probably plenty of astroturfing out there, but I don't think this is a strong sign a particular account is astroturf. Since we are all exposed to everyone's discourse, that way of referring to Democrats could fairly easily hop between different groups that don't like Democrats. Like I wouldn't be surprised if "scratch a liberal and a fascist bleeds" makes its way to the right somehow just because it's anti-liberal, even though it originates from the left.
Despite its origin on the right, I think that type of phrasing makes a bit more sense coming from the left than the right, because the right does not actually care at all about democracy, except when they equate it with capitalism, and democracy is like the left's whole deal.