I can't help but imagines the Moon Presence giving Dreamer a headpats while the Dreamer just standing there dumbfounded (maybe fear too) at the Eldrich beings showing its "affectionate" .
Knowing that the Great Ones all yearn to have children but lose them at birth makes it understandable, somewhat, why the Moon Presence desperately seeks out Dreamer. It fully believes Dreamer is their child separated at birth. Dreamer of course, is reasonably afraid of the Moon Presence, especially with how, once reader becomes aware of the Moon Presence, it overwhelms and takes over seemingly every aspect of them.
I don't exactly think the Moon Presence is capable of being "affectionate" in a traditional sense. It's this eldritch being that exists beyond dimensions, it can't ask parental in any traditional way, which is why it acts through Gehrman, its original surrogate, to interact with Dreamer.