Is halo 3 multiplayer separate from halo 3?

No, the Halo 3 game disk has both the campaign and multiplayer mode on it.

What might seen confusing is in Halo: ODST, the game comes with two disks. One disk is for the Halo: ODST game and the second disk is for the Halo 3 multiplayer with all the latest map packs on it such as Mythic