It's not ACLU's identity crisis, it's the Left's identity crisis. Civil rights are checks on government and institutional powers. When the left gained a massive institutional power and a lot of governmental power, it turned out it doesn't really like this power to be checked and challenged. It's one thing to fight for the free speech of people who are maybe abhorrent, but ultimately their speech won't change much, it…
It's easy to be principled when it doesn't matter. To anyone who didn't live through it, the Nazis are an ancient demon that feels as relevant as Darth Vader or the Devil. What do I care if 4chan wants to name a soda "Hitler did nothing wrong" for the lols?
It's a lot harder when you perceive the speech that your principles would have you save to be a tangible threat to your safety. If some on the left (rightly or wrongly) see e.g. unmoderated alt-right social media activity as a risk factor for something bigger than 1/6, there would be an inherent tension between principles and pragmatism.
I'd be interested in seeing evidence as to whether the left has trended toward a particular side, but it's a shame that this has to become an issue in the first place.