Is not being paid in an internship common in academia?
Sadly, yes. It is all too common, though not universal. Not only for internships in labs, but also in businesses. The US Congress, even, doesn't pay its interns, though there is some move to do so in future. 

The "justification" of this is that the students are "learning", which can be true or false. But normally they do work for which pay would otherwise be required. 

In my opinion, this is just wrong. If people are contributing, they should be paid. If they are not paid then many people who need to support themselves - somehow - will be closed out of these opportunities. I would have a bit less objection to it but for this fact. In economically diverse countries such as the US (and others) it is especially important. 

The solution to it is political, however, whether at the university level or the governmental level. 

As you say, of course, an internship teaches you things that are hard to learn otherwise. They are valuable for themselves. Your CV can reflect internships you have done. 

Well, some internships teach you things. There are also examples in which "interns" become bottle washers, not learning much of anything. This is pure exploitation. 

But labs, agencies, and businesses need to include sufficient funding so that students don't get exploited.