The ACLU is much more closely aligned with the Libertarian Party, although they don't officially associate with any political party. They were huge critics of the Obama administration, the "embodiment of the Democratic Party" -- to some people, at least. They've also been huge advocates for Title IX reform on college campuses and opposed restrictions of student expression. It's wishful thinking that they are any closer to the Democratic Party than the Republican Party -- both are enemies of freedom, most of the time.
I don't think the ACLU is nearly as partisan aligned or affiliated like that. But they're definitely more aligned, as a political matter, with the Democratic party. Partly for historical reasons (e.g. school busing, voting rights, criminal rights, etc.) But I totally get what you're saying.
The other part of it, though, is that we all need to learn to be a little more tolerant of political organizations and politicians taking stances that we personally disagree with. It's inevitable. To demand otherwise leads to more extremism, and leads to more fragmentation. If all we have are 10,000 smaller advocacy organizations, they're all going to drift left or right to a much greater extent. The only way to arrest that shift is to use yourself as an anchor; support the organization but then be sure to voice your opposition when they assume an inappropriate policy stance or affiliation.