For Settlers II: 10th Anniversary, I was looking under the More button on the game shelf and saw the “Serial Keys” option. But when I clicked on it, it give me the error “Serial keys depleted.” I've attached a screenshot. Does that mean that multiplayer will never work for this game? Does the game even need a serial key?

At least for the LAN mode you don't need a key. I have played this game many times without needing one. I don't know about online multiplayer (if that even still works) though.

From store page:
"Multiplayer notice: The game's multiplayer servers have been taken offline, however, multiplayer over LAN is still available."
And for LAN mode you don`t need the key (see post above).