Does it? I can think of any number of civil liberties which I would associate with the "right" wing of American politics. Right to bear arms, for example, freedom of speech and religion, right to life. The "stand your ground" phenomenon, a very American thing, is definitely right-wing.
It seems to me more that the "right" and "left" have adopted certain civil liberties for their pet causes, and the ACLU, a left wing organisation for sure, advocates for their favoured issues. But let's not confuse that specific organisation and their agenda with the general concept of "civil liberties" per se.