Is there anything like a server ID, kinda like a game ID but for a certain server?

If so, is there a way to teleport players to it, kinda like teleporting players to a different game?

Of course! Every server will have a randomly generated ID. You can get it with the game's JobId property.

print(game.JobId)

Oh and for the teleportation, you can use TeleportToPlaceInstance,

game:GetService("TeleportService"):TeleportToPlaceInstance(placeId, jobId, player)