"Театр находится на другом берегу реки."
Translation:The theater is on the other bank of the river.
Technically it's the locative case. Some nouns in Russian have a locative case. A leftover from older forms of Russian that's only used with prepositions of position.
I translated it as "The theatre is on the other river bank." It was counted incorrect, saying it should have been "the other bank of the river". Is the first actually incorrect; they mean the same thing in English, but perhaps they translate differently in Russian.
Why I can't translate «берег» as «coast»? What the difference btw «coast» and «bank»?
This is an English issue. Rivers have banks, oceans and seas have coasts. Some very large lakes have coasts too. If you want an English word that can be applied to all bodies of water the way "берег" can, use "shore."