Hestia is the goddess of the hearth, the focal flame of both private dwellings and communities, yet She’s not the only one to play the role of house deity: Zeus Herkeios, for instance, protects the building and everything in it and often had an altar in the courtyard; another domestic aspect of the King of Olympus is that of Zeus Ktesios, where He appears in the form of a snake; Apollon is the guardian of gates and doorways or, to quote a post by Lykeia, He’s “the protector of the inviolable nature of the oikos“; the Dioskouri, who are well known as gods of horses, warriors and sailors, were also worshipped as house gods, especially in Their native Sparta. What these examples show is that a deity doesn’t have to have a domestic origin to be a domestic Power – if He/She’s important for the family as a whole and its dwelling, He/She can play a role as a household god/dess. So if Ingui provides food, heirs and protection of both people and animals, if He bless one’s home as a place of peace and inviolability and has two probable house elves as servants, it seems clear to me that He can be a god of the household. The connection is hinted at by His own name/title: He’s Frey or Frö, the Lord or Germanic pater familias; and in a modern Latinized cult the term can be translated as Dominus, the master of the house or domus.
Does this mean that Ingui is a god of everything? The answer is obviously no, though I understand why some may ask that. But the thing with polytheistic deities is that They don’t usually fall within clear-cut categories with very well defined “functions”. There’s a large degree of overlapping and a natural development of roles from a core nature: Zeus is a celestial father, so that extends His role of guardian and judge to the microcosm that is a home, where there is also a father, rules and a need for protection; Apollon is a god of light with an apotropaic nature and that naturally places Him at the gates, protecting the dwelling and the flame within the building.
Similar elements of relevance for domestic life and its survival can be found in Ingui as well and that, once the connection is established, places Him in the role of a god of the home.