Fungus, not necessarily a mushroom. Yeasts and molds are also fungi.
He's probably referring to a future strain of Fusarium venenatum. A current strain is now being produced commercially.
It makes sense if your space and power is limited. It's easier to make a single source of protein and flavor it, rather than have multiple cloning facilities for all the meat types.
And, as it's a simpler genetic structure, it'd be easier to engineer necessary nutrients into it. Various amino acids, enzyme production, etc.