Dough isn’t just for bread, the word dough is used for cookies too.
Batter is for cakes. As people said above, batter is more liquid than dough. Also, batter can be used to coat fish, like fish sticks, and also tempura veggies. Fish sticks can also be bread crumb coated, that’s something else.
The words in English are mostly based on how thick or liquid, and also common usage. I make a pound cake that has a very thick batter, it might be confused with dough if someone just looked at a bowl of the stuff, but it’s customary to say cake batter.
Other languages only have one word for both, so in translation it can be difficult for people using English as a second language.