The core name is 'Moonphaze' — a stylized spelling of 'moon phase,' matching the brand's moon-and-nature theme. That is the spelling on the Google business profile and on review listings, and it is also the spelling of the unrelated 2021 music track.
Two extended spellings appear as well. The brand's Instagram handle is @moonphazeee, with three trailing e's — a common tactic when the plain handle is taken — and this reference site uses moonphazee.com, with two. Searchers should expect all three spellings to surface mixed results covering both the cafe and the song.