There isn't a specific term for an organism where all offspring across many generations have the same trait. This is because it's very unlikely in nature.

Here's why:

* Genetic Variation: All sexually reproducing organisms have genetic variation. This means their offspring inherit a mix of genes from both parents, leading to differences in traits. Even if a trait is dominant, there's always a chance of recessive genes being passed down, leading to variations.

* Mutations: Mutations are changes in DNA that can occur spontaneously. These mutations can introduce new traits into the population, leading to further variation.

* Environmental Factors: The environment plays a significant role in how traits are expressed. Even if an organism inherits a trait, environmental factors like diet, climate, or disease can influence how that trait manifests.

What terms might be relevant?

* Purebred: This term applies to domesticated animals where breeders have selectively bred individuals to maintain a specific trait for many generations. While this minimizes variation, it's still not guaranteed that all offspring will have the exact same trait due to the possibility of mutations and environmental influences.

* Clones: A clone is a genetically identical copy of an organism. Clones would have the same traits as their parent, but they aren't usually produced over many generations.

* Species with low genetic diversity: Some species have very low genetic diversity, meaning there is less variation within the population. This can make it more likely that most individuals will share a particular trait.

Important Note: In real-world situations, even in purebred animals or species with low genetic diversity, it's highly unusual for all offspring across many generations to have *exactly* the same trait.