A single player game with a co-op mode is not an MMO.
Streets of Rage has co-op. Streets of Rage was not an MMO.
Now, in this example, you are playing with the other person, not against them. You are not competing with them, and so it makes no difference if they have some better items, since that just means they will be better at helping you survive the dungeon.
In an MMO, you are competing with the other players, which is why buying items for real money is bad. PVP and PVE both involve competing with the other players who are also playing, either directly against them in PVP or for things like world first kills in PVE.