Is publishing as an undergraduate a thing that really happens?
Most undergraduates do not publish, but it's quite normal for undergraduates who get involved in research to end up publishing.

Reasons why few undergraduates publish include:


They aren't interested in research, or are more interested in all the myriad other awesome things you can spend your time on as a newly (semi)independent adult in college.
They are interested in research, but don't manage to find a good mentor (typically a professor) to work with.  Nearly all undergraduate institutions have professors doing research, but some institutions or departments within an institution have a lot more going on than others.
They get connected to a research project, but their work isn't publishable.  Undergraduates are typically unreliable (see first point), and so often are given high-risk or boring work (a friend of mine got a "research" job one summer crushing rocks into fine powder with a mortar and pestle).
They do publishable research, but it doesn't come out until they are no longer an undergraduate.  Even the fastest project rarely takes less than six months from start to publication, and many take multiple years. If a student starts researching in their junior or senior year, even a great project may not result in publications during undergraduate.


That said, none of these are particularly insurmountable obstacles, and with the right combination of interest, a good environment, and a bit of luck, it's quite reasonable to have work published as an undergraduate. 

Some undergraduates even become quite well-published, simply by virtue of having the right combination of skill, luck, and circumstances.  For example, this week a colleague of mine mentioned to me that one of his undergraduate students now had approximately a dozen journal publications. Until they said it, I hadn't realized this student's numbers had accumulated quite so much, but it didn't surprise me, since this student is a good contributor to a complex many-person project that publishes frequently.

In short: undergraduate publication is statistically infrequent, but entirely normal.