Re SAME genetics -
It's not that simple.
If a bird is homozygous for a given allele, say "c" (recessive white, in Tara's example), it implies simply that, i.e. that she's got two copies of "c". It says nothing about homozygosity or heterozygosity at all those other loci. At some (quite possibly far remote) level, that means she's somewhat (possibly only a tiny bit) inbred, since that allele originated at some time in a long-ago bird by mutation from the wild-type C. Barring multiple mutation events with the same genetic implications/outcome (possible but not the norm, and they don't really change the logic here) she must have inherited both copies from that ancestral bird. However, many generations of random mating and recombination (or selection and breeding, also accompanied by recombination) could lead to her achieving two copies of that allele, either by chance or design, with no matings between near relatives in her recent history. The important thing is that her alleles at other loci could have been inherited from any of her gazillions of ancestors, not necessarily the one with the "c" mutation.
Non-wild-type alleles that are identical at a given locus (homozygous) are called IBD (identical by descent).
If a bird is inbred, it increases the likelihood for each locus of being IBD, not just the genes/loci of interest. And therein lies the trick of breeding - to use breeding as a tool to get the loci you care about, i.e. those that define the breed, to be IBD, while maintaining many others (especially those that are fitness-related) in a state of (generally) wild-type homozygosity.