Catherine is absolutely full of mythological references, most of them Babylonian and Sumerian.
Ishtar is the central figure in the story, the one behind everything. She is the Babylonian Goddess of fertility, love, war and sex. Her Sumerian form is Inanna, Sumerian Goddess of sexual love, fertility and warfare. It is important to note in all her forms, Ishtar is not a goddess of love in the romantic sense, but purely in the lustful and physical sense.
She is often associated with the planet Venus.
Dumuzid is another figure in Sumerian tales. Early tales call him Dumuzid the Shepherd, a king that ruled Sumer for 36,000 years. Later, tales speak of him as a god. This transformation in myths is mirrored in the game, as Dumuzid was previously mortal. Dumuzid is considered the precursor to Tammuz, the Sumerian god of food and vegetation.
In both Sumerian myth and the game, Dumuzid is Ishtar's long-term consort: effectively, her husband.
The tale he is best known for is his banishment to the Underworld by Ishtar. After she escaped from the Underworld under the condition that she find a replacement soul, she finds that he is the only one not mourning her (in the Rapunzel game that tells the tale, he is actively sleeping with other women). Disgusted, she commands the demons to take him to the Underworld in her place.
This falling out mirrors their in-game relationship, where she is sick of his behavior and throws him out.
Nergal portrays Catherine's father in the game. He too is a Babylonian god of war and pestilence, and is known as the lord of the Underworld with Ereshkigal, Ishtar's sister. He is often depicted with the head of a lion, which can explain his very bushy beard/mane in the game. He has also been associated with demons and Satan due to his aspect of fire.
He is often associated with the planet Mars.
In the mini-game Rapunzel, the entirety of the 'Extra Mode' is a retelling of the myth "Ishtar's Descent to the Underworld". Though some details change (mostly to paint a more sexualized tale), the base story is the same.
KRIOS is the company that manufactures Vincent's cell phone, provides the service, and is credited as programming the Rapunzel arcade machine. Given all these elements are linked to The Great Trials (the cell phone is wiped of Catherine's information, Boss placed the Rapunzel cabinet in the bar), the company can be assumed to be quite suspicious.
KRIOS only shows up in Greek mythology, a sharp departure from the rest of the game. Krios (Crius) was a Titan, and with his brothers assaulted their father, Ouranos.
Interestingly, he is considered to represent the southern cosmic pillar (the Rapunzel re-telling of 'Ishtar's Descent to the Underworld' is described as a song from a foreign land to the south), and is represented by the constellation Aries: the Ram. Κριως (Kriôs), in Greek, means 'Ram'.