Replace prop=revisions
with prop=categories
and you will get a list of categories the article is in.
For example, api.php?action=query&prop=categories&titles=pizza
gives you (among other things):
<api>
<query>
<pages>
<page pageid="24768" ns="0" title="Pizza">
<categories>
<cl ns="14" title="Category:All articles needing additional references"/>
<cl ns="14" title="Category:All articles with unsourced statements"/>
<cl ns="14" title="Category:Articles including recorded pronunciations"/>
<cl ns="14" title="Category:Articles needing additional references from June 2010"/>
<cl ns="14" title="Category:Articles with unsourced statements from March 2013"/>
<cl ns="14" title="Category:Flatbreads"/>
<cl ns="14" title="Category:Greek inventions"/>
<cl ns="14" title="Category:Italian cuisine"/>
<cl ns="14" title="Category:Italian inventions"/>
<cl ns="14" title="Category:Mediterranean cuisine"/>
</categories>
</page>
</pages>
</query>
</api>
Note that you get the article categories, not some kind of taxonomy of the thing the article is about. You could probably follow a 'chain' (more like a tree) of categories up to a handful of large categories that you care about. For example, the Flatbreads category leads to Breads, which leads to Foods. This means that you could assume that pizza is a food. It's no guarantee, though. Article categories are not really meant to be used that way.
If you want a better classification of 'things', instead of articles, try something like Freebase - which, for example, helpfully lists pizza as a kind of food. A sample query would look like this:
[{"id":null,"name":"Pizza","type":[{"id":null,"name":null}]}]
This returns, among other things, Food, Cuisine, Bread and Man-made Thing.