What is it called when you classify words into categories, but not by part of speech? I'm looking to classify words into type categories like sushi to food, baseball to sports, happy to emotions, panda to animals etc.

What is this type of classification called? I see lots of NLP-related articles discussing parts-of-speech classifiers. This isn't helpful.

I know any word has many possible categories. For example, panda could be classified as animal, mammal, Ailuropoda (it's genus), or even algorithms (as in, "Google Panda"). I'm open to any of the above to start. My preference would be colloquial rather than scientific (panda to animal).

Ideally it would return probability for different categories. For example, panda might return 0.8 for animal and 0.3 for bear and 0.1 for algorithm.
The name of the classification you are looking for is called Semantic field classification (https://en.wikipedia.org/wiki/Semantic_field). The sort of algorithm you are looking for falls under the category of Statistical Semantic.