Yes, they're both sweets where I'm from.

[At least it communicates clearly to him that the food must be foreign to her]

Daifuku is made with mochi and sweet bean filling. Mochi is rice that's been pounded into a kind of springy paste and then molded into the shape you want. Taiyaki's a cake shaped like a fish, with something like chocolate, custard, or sweet bean filling. You use pancake or waffle batter for the cake, and it's baked in a mold.