Why Does Stockfish Make Bad Moves Sometimes When Losing, When Analysis Board Suggests a Better Move?

When analyzing where I have a definite and overwhelming endgame winning position, sometimes Stockfish doesn't take a piece I leave hanging, or it moves away from protecting a square that needs protection, or some other weirdness. These are situations where I am winning regardless, and if I turn Stockfish off, the analyze board suggests a better move, although it still isn't good enough to change the outcome of the game.

Does Stockfish just give up and play badly when it is in a badly losing endgame? Is it playing with the idea to take advantage if the opponent blunders in some way it predicts might happen as it's only chance to win or draw? Or what is going on there?

The human brain is "designed" to recognize patterns. Various Stockfish levels introduce more or less randomness into the moves. You're not alone in thinking there are patterns in randomness.

Of course lower stockfish levels are designed to make mistakes so is this for SF8 you are tlaking about?
Also, sometimes a move that looks bad, such as not taking a hanging piece will result in a longer distance to mate than by taking it. For example because by taking it it puts the king into a mating net.