There is a nearest-to-meaning amongst these answers but not truly a synonym. Purebred is a popular but unscientific word, but it gives the idea of a lack of influence from other allelles. Zygosity is the degree of similarity of the alleles for a trait in an organism.
To by homozygous the alleles will be similar at corresponding chromosomal positions in order to produce the trait - such as red-eyed - for we are talking of all species here, not just humans. Zygosity pertains to the study of Genetics, which began as a science by the study of fruit flies.
Basically, a homozygous cell or organism has two of the same form of a particular gene for a set trait.