What is a word that means an instance of a game?

Solution 1:

From the Free Dictionary:

game n.
2. a. A competitive activity or sport in which players contend with each other according to a set of rules: the game of basketball; the game of gin rummy.
b. A single instance of such an activity: We lost the first game.

To indicate in progress just use present continuous tense:

"We can't come outside right now, we're playing a game of Monopoly."

Solution 2:

A round is one word that is used.

For those unfamiliar with the use of round, see:

Oxford English Dictionary

Round
VII. A period or bout of play at a game or sport, and related senses.
34. a. summary: a game of cards = a round of cards
b. summary: refers to a round (=game) of golf
c. gen. A bout of play at a game, sport, or contest; (in early use) spec. a bout of fisticuffs. Also in extended use.

1962 E. Albee Who's Afraid of Virginia Woolf? (1965) ii. 85 We'll play a round of Get the Guests. How about that? How about a little game of Get the Guests?

1994 Rotarian Sept. 4/1 Planning a visit to a sister club in the Philippines, and playing a round of mah jong, were far from regular, bland tourist staples.

2009 M. Herczog Frommer's New Orleans 2009 xi. 277 Enjoy some shepherd's pie, wash it down with a sample of the nice selection of beers on tap, play a round of pool or darts.

THUS: a round of Monopoly

You may not say it like that, but me and my kind do.

Solution 3:

We distinguish by using the definite article to refer to the concept and "a game of" to refer to a round or marathon session of Monopoly:

Who likes to play the game Monopoly?

Who would like to play a game of Monopoly?

Solution 4:

you could refer to a game instance as a match and also a friendly