May be misunderstanding, but I am not sure that the concept that you are talking about actually exists. Recombination in each generation will break up contiguous chromosomal sequences into shorter regions that are passed down each generation. Furthermore, such a familial genome has an artificial bound, as ultimately all humans are related in a single pedigree due to the likely single origin of humankind.
Some related concepts are:
- pedigrees
- trios (parents and child genomes sequenced, usually to track where a child's genetic disease comes from)
- quads (parents and two child genomes sequenced, usually where one child is affected by some genetic disease- no convenient link but similar to trios)
The closest thing to what you are talking about is probably the haplotypes shared by all family members. Haplotypes are the regions of sequence that can be shared by families that are not broken by recombination. You could speak of the set of common haplotypes between some set of people (in a family or just in a population).
However, haplotypes will necessarily get chewed up by recombination over time, so they're more of a heuristic for talking about the set of shared genotypes in a group of people (who are after all still related in some big pedigree), not something that's specific to lineal descent of families such as you suggest.