While fealty represents the great house that holds authority over a character, 'family' represents the immediate house, clan or family of which a character is currently a member. This does not necessarily have to be a blood relationship- wards may be adopted and take the surname of the family they join, or remain as members of their own house despite living among others. Likewise, marriages typically (but not always) involve one of the parties joining the other house as part of the marriage. The culture on marriages varies from house to house, with only House Thrax and all its vassal houses maintaining male primacy without exception.
Pre-made roster characters will have their family defined and typically occupy an important niche in that existing family, with a clear lineage and role, to allow new players of a roster character to already have a significant social network they can roleplay and interact with. For characters generated by players themselves, membership in existing families must be approved and might require modification of a character's lineage or backstory to make work, but we are extremely happy to help with that. Further, players are very much encouraged to create their own families, and the noble houses that act as vassals to the great houses are left intentional vague just so players can design entirely new noble houses with their friends if they are feeling ambitious. If anyone is willing to go to that degree of trouble, of course staff will be equal parts surprised and ecstatic to help make it work into the lore of the game.
For a list of families, see 'help families list'