Is classification only a machine learning problem? Wikipedia says that:

In machine learning, multiclass or multinomial classification is the problem of classifying instances into one of three or more classes.

It looks like the problem is strictly related to machine learning. But to me, this is just a problem to solve and machine learning is only one of the possible approaches to solve it. And I can compare different approaches using same metrics because the problem remains the same.
What are other approaches to multiclass classification than machine learning?
I implicitly assumed there are because I approached one of such problems before I even knew machine learning exists. I had to classify products into shop departments they can be located in. I labeled manually a few products and inferred labels for the remaining ones based on products taxonomy, e.g.: Jack Daniel's and Johnnie Walker's are both whiskies, so if I know the first one can be located in the Alcohols department, then I can guess the latter as well. A very simple approach that just traverses the taxonomy graph without using machine learning at all.

You are exactly right. Machine/statistical learning is one approach to classification, but not the only one. Simple rules created by humans are probably more common in computer programs than ones created by ML.
