It could be both singular or plural, depending on context. Peter plays that game पीटर वह खेल खेलता है। Peter plays the game पीटर खेल खेलता है। Peter plays games पीटर खेल खेलता है। Is Peter playing with me? क्या पीटर मेरे साथ खेल रहा है? Is Peter playing games with me? क्या पीटर मेरे साथ खेल खेल रहा है? Is Peter playing? पीटर क्या खेल रहा है? What is Peter playing? पीटर क्या खेल रहा है?
Consider this: the article 'the' was not a word available among choices to form a solution, so actually the alternative was between two answers: Peter plays game, Peter plays games; the sentence with the plural makes perfect sense in English, meaning Peter usually plays games, whilst the singular Peter plays game is not a correct sentence, means nothing, for it to mean something you must add either 'a' or 'the' in front of 'game', but you do not have any of them. The only correct meaningful answer that you can build out of available English words is 'Peter plays games', so 'Peter plays game' should not be accepted.
You are correct that ''Peter plays games' is the only correct response you can construct out of the word bank. However, a lot of people use the keyboard instead of the word bank to type their own answers. You can switch between the two on the desktop version of Duolingo and on the app version, the higher levels use the keyboard.
Namaste! I think the verb declension is right. The Hindi verb खेलता is masculine and singular. And, the direct declension of the noun, singular or plural, is खेल. But, the Hindi verb खेलते is plural.
So, I think, the sentence is:
पीटर (Peter) खेल (games) खेलता है (plays) | / Peter plays games.
Verb declension of खेलना (khelnā, “to play”), from Wiktionary: