They describe a leaderboard built from server members' shared gaming activity, while actual rollout remains unconfirmed.
When It Appears
“{count, plural, one {# game} other {# games}}” can appear on a server view that ranks shared gaming activity and shows a timed competition when Discord displays the plural-aware counter for displaying a number of games.
When “{count, plural, one {# game} other {# games}}” is shown, the current step is presenting the plural-aware counter for displaying a number of games.
Search Intent
A user who saw “{count, plural, one {# game} other {# games}}” in Discord is likely searching to confirm that it means the plural-aware counter for displaying a number of games, where it appears, and which state it describes.