What I'm asking about is kind of architecture seen (more or less) in Minecraft, UT (at least the first one and its' mods), Payday 1, L4D 1 and 2 and similar games, where Singleplayer is just Multiplayer with both client and server being on localhost - yes, there are many technical differences between the modes even if bulk of code is shared and it's not as simple as that, but the general principle is that SP is sort of MP with bots or just player alone.

Is there a specific name for this kind of design?

AFAIK this originated with the original Quake engine, and is what virtually every FPS since has modeled it's netcode after.

AFAIK this originated with the original Quake engine, and is what virtually every FPS since has modeled it's netcode after.