"We're a duopoly not a monopoly" is not a particularly compelling argument.
In particular, "monopoly power" is typically not interpreted (in the US or EU) to mean literally a single supplier but rather whether you exert dominance in the market. You're correct that "it's a duopoly" is not a compelling argument but Apple would not bother to try to make that case either. They'll just point to the fact that they have about a quarter of the market world wide and slightly more than that in the EU alone; evening the US it's a little less than half the market.
I suspect they have a strong case legally, whatever the merits of Spotify's claim, and doubt they would prevail in court (though IANAL -- it doesn't matter what I think). However if the lawsuit is part of a strategy to get the EU competition commission to act, then it can be effective, even perhaps more effective if they lose!
I believe Spotify is claiming Apple is doing this with Apple Music vs Spotify and I didn't see this issue addressed by Apple's message.
I am not sure I agree with Spotify's claim, and US antitrust is in a parlous state, but this approach may get traction in the EU. However Apple's market share is far far from monopolistic.