What vocabulary term refers to the offspring when it possessing the same trait for both genes?

Homozygous

Explanation:

Homozygous is a word that refers to a particular gene which possesses identical alleles on both homologous chromosomes. It is referred to by two capital letters (YY) for a trait that is dominant, and two lowercase letters (yy) for a trait that is recessive.

Therefore, if an organism possesses two copies of the same allele, for example AA or aa, then that means it is homozygous for that particular trait.