After his victory over the first strongest monster, the Prime Stone Goliath, Adam would take something from the monsters he defeated to add to his collection - a trophy. This applied to some other important memories as well, but mostly monsters, because he was a Phantom.
He fought against monsters. That's how he survived, and that's how he became stronger.
Even though the Crimson Monarch was his greatest opponent up to that point, Adam couldn't get anything out of him. The Crimson Monarch died without leaving anything behind except Moira.
Adam couldn't put Moira in his storage, not because he didn't want to, but because she didn't have a physical body.
He didn't think about it too much, considering that Moira was to the Crimson Monarch what Lone Blaze was to him.
On the interaction side, he wasn't wrong, they helped each other, and shared energy and abilities, just like Phantom and the External Artifact.
