The Last of Us 1 does have multiplayer, although it depends on which version of the game you're playing. If you're playing The Last of Us: Part I on PS5 or PC, then there is no multiplayer mode. This is because Naughty Dog is working on a new, large-scale standalone multiplayer game which was announced alongside The Last of Us 2. While details are scarce at this stage, more information is expected in 2023.
If you're playing The Last of Us Remastered on PS4 or The Last of Us on PS3, however, there is a multiplayer mode named Factions. This retains the core gameplay loop from the main campaign, requiring you to scavenge for supplies as you seek to win matches to build up your camp. It offers an extremely unique competitive shooter experience, and is well worth checking out.