Randombred cats can have all eye colours. When a random bred cat has green eyes it doesn't have to be a sign of purebred parents.
In some breeds certain coat colours need to have certain eye colours and that is achieved through selection, there is no "green eye gene" that is a sign of a certain breed.
In cats you can have blue eyes (white/bicolor and pointed cats, with a special colour in mink cats) but most cats have colored eyes, black cats too.
Colored eyes can have different colors based on how densely they are pigmented. Copper eyes have the most pigment, orange eues a little less, then there is yellow and finally green eyes have the least dense pigmentation.
With blue eyed cats you can see this a little too, blue eyes are not really pigmented but they have fragments that reflecet the light a certain way. A lot of fragments = light blue eyes, few fragments = deep blue eyes.
How densely eyes are pigmented can be selected for, in Persians and British shorthairs for example they select for deep copper eyes in the self colors and for green eyes in the silvers. In Siamese and Birmans they select for deep blue eyes. The sister breed of the Siamese, the Oriental shorthair has green eyes. That is logical because deep blue and green both have to do with very little pigment.
I have a randombred pointed cat with pale blue eyes and a Birman with deep blue eyes. If the Birman would be a non-pointed cat his eyes would be blue and when the Domestic Longhair would be a non-pointed cat she would probably have the same eye-color as her littermate who has yellow eyes.
The eye color of a kitten can change a lot later on in life.
This is my cat Flynn at 10 months of age, he has orange eyes;
Here he is 3 years old;
7 years old;