Is the word game a common noun?

Yes, the Word 'game' is a common noun, a Word for any game of any kind; for example:

• checkers
• Video Games
• solitaire
• chess
• poker
• video poker

A proper noun is the name of a person, place, thing, or a title; proper nouns for games are the 'brand' names of games; for example:

• Angry Birds
• Animal Crossing
• Monopoly
• Naruto
• Scattergories
• Scrabble