They describe a leaderboard built from server members' shared gaming activity, while actual rollout remains unconfirmed.
When It Appears
“Rank {rank}” can appear on a server view that ranks shared gaming activity and shows a timed competition when Discord displays the label displaying a user's numeric leaderboard rank.
What It Means
When “Rank {rank}” is shown, the current step is presenting the label displaying a user's numeric leaderboard rank.
A user who saw “Rank {rank}” in Discord is likely searching to confirm that it means the label displaying a user's numeric leaderboard rank, where it appears, and which state it describes.