I had my toby fever dream moment last night
I woke up with an extremely vivid assurance that Gaster is Susie's dad, and her full name is Susie Gaster. More specifically, "The Susiegaster", in her dragon form, which is a pun on the Snallygaster which is a lake-dwelling tentacled dragon cryptid from my home state of Maryland.

This connects to the underwater caves, the lake and the old song from the sea. It's her home.