Is it possible to categorize the product tags that are attached to the products?
For instance, for a food product that is tagged with tags like 'Asian' and 'spicy' or 'Italian' and 'sweet', it could be helpful for customers if these tags where sorted by the headings 'Country' and 'Taste' in the sidebar - does anyone know if this is possible?
Yes, it's possible you check in Empire theme. Just search for it and in its demo check collection page you will find how in sidebar tags are working.
Note: I'm not advertising theme, just use the theme name to show an example.