If the restaurant said "birthday reward when paying the bill", I could see that as possibly meaning a discount -- by time you pay the bill, surely you've ordered dessert already if you'd wanted it? But yes, my accumulated restaurant experience says "probably a generic dessert, possibly brought out loudly by the staff".

(It now occurs to me, though, that at least one of my kids may never have seen the big fuss dessert practice, at least when at an age to remember it, and at the rate things are going with Covid and inflation, they might not see it until they're an adult.)