Wikipedia just says it's "made with gin and blackthorn fruits (sloes)"

Sloe gin's a bit more complicated - it's technically a liqueur, with the sloe berries and sometimes other spices or sugar added to the gin after distillation.

oh! So it is added later. I've seen people distill alcohol from the sloes, too.

That seems like it's a different booze, like mead vs Drambuie even though both use honey

That's a good point.