Player 1 hosts a world. Player 2 joins via LAN.

Player 2 wanted to play alone and logged in as themself on Player 1 laptop (where the game is saved).

Player 2 appeared in the game with the appropriate skin but with several incorrect attributes:
1) advancements were for Player 1
2) inventory was Player 1
3) Ender chest was Player 1
4) Location was Player 1

Shutting the game and opening as Player 1 then letting Player 2 join via LAN gives the following situation:
1) both players have Player 1 advancements
2) Ender chest have the same contents for each player (i.e. Player 1 ender chest contents are also in player 2 ender chest)
3) same story with location.

It seems to me that the game should be able to track inventories etc. from the multiplayer game in single player mode.