Exactly.
He is the first Ultraman on Earth.
When Zoffy showed up at the end of his series, that immediately rendered the name Ultraman to a title.
So, he needs his own name that matches both his historic mark on humankind, and the fact that he is the field leader of the Ultramen. Something that gives it the appropriate weight and dignity. Simply using an adjective and Ultraman doesn't work as well as having an independant name.
Hence, "Prime". Says first and foremost, since he's usually the leader of the Ultra Brothers when they come to Earth.
Admittedly, the idea of his adopting his host's name is rather amusing. Almost as if some sort of more formal relationship had been established besides a mutually- beneficial life-force bond. (Comrades? A temporary adoption? Making them "blood brothers"?) But, no one else is referred to by their host's name. And besides, it's a bit prosaic. Can you imagine Tiga as "Ultraman Mandoka"? Or Dyna as "Ultraman Asuka"?* Or Gaia, as "Ultraman Takayama"?
Besides, everybody is getting names as quickly as possible. We now know what the names of the Father and Mother of Ultra are. We may find out what Ultraman King's name is in the future.
*I wonder what name Shin Asuka was given when he arrived on Ultra? :eyebrow: