Monopoly here is legal jargon for market power. Apple doesn't need to literally hold the deed to the three phone properties, which would entitle them to charge twice the rent or build houses and hotels.
The global market provides interesting background information, but also doesn't have much weight for US anti-trust.
In the roughly 60/40 market that we have in the US, both OS developers likely exhibit strong market power and their related conduct is up for review. Do they use their market power in mobile phones to further distort the market for mobile phones in prohibited ways? Do they use their market power in mobile phones to distort other adjacent markets in prohibited ways?
Personally, I don't think DoJ is going to have a problem showing market power. It's clear that Apple has market power in many countries, including the US. There certainly are many countries where it doesn't, but lack of global market power isn't very relevant.
I don't know whether DoJ will be able to show abuse of the market power. IMHO, the best of the five examples they presented is smartwatches; did they establish dominance in the smartwatch category by limiting deep integration to only their smartwatches, and is that prohibited tying? But we'll see how that holds up at trial.
Cases against Google are more complex because Android allows significant manufacturer customization, so Android the OS has 40% market share, but Google the seller of phones has much less. And Google allows software developers to do a lot more with apps, for better and worse.