>How is race a discrete, and not a continuous, property?

The "one drop of blood" school of thought.[1]

[1] https://en.wikipedia.org/wiki/One-drop_rule

'Race' is a thought-stopping way to categorize humans, but if we replace it with 'ancestry' rather than 'Census designation category' we can at least state with confidence that ancestry is a discrete property, one is simply descended from another person or one isn't.

It doubles with each generation (this stops being true remarkably quickly but let's use single-static-assignment to ignore cousin marriage), so while someone with perfect information could say "so-and-so has precisely 14 ancestors from West Africa in a generation where the remaining 242 ancestors are all European", it's easier to say they're 5% African-descended.

Or Black, by the old American rules around race. I did say it's a thought-stopping category.

Race isn't a discrete category. Ethnicity is closer to what you're describing.