The same issue can be a civil rights issue if it's engaged with seriously by people who actually care about those it affects and a culture war issue if it's engaged with unseriously by hatred-blinded idiots who just want to own the libs or own the deplorables.
But "civil rights" is just term du jour for idiot libs who want to own the deplorables to couch their arguments in sanctimonious appeals to some normative quality.
Both of these phrases are absolutely meaningless. It has nothing to do with how "serious" those using them are.
I'm tempted to make a snide remark about a R-MS avatar saying that "civil rights" is a meaningless phrase, but I can't quite put my finger on one that wouldn't just be a pat insult, so instead I'll take this conversation seriously and say that just because the phrase "civil rights issue" is overused doesn't mean that, say, employment nondiscrimination and police brutality aren't actually civil rights issues. They clearly are. If you're referring to post-
Obergefell small-ball LGBT issues like the Bathroom Question and the Wedding Cake Question, or to the claim that (say) positive economic rights to things like publicly provided food and housing are "civil rights issues", then sure, you have a point.