How to filter an already rendered calendar view of a specific date with categories (taxonomy terms)?
If your "list of every event on that date" is generated by Views, you can add a Filter to Views based on Taxonomy Term.  This can then either be exposed to the user so they can pick Movie or Sport - or you can select which one you want explicitly for them.

See selection options below - you'd likely want "Taxonomy: Term" (and you'd be doing this in the Filter - not Contextual Filter).