I think I understand you not sure though correct me if I'm wrong:-)
You would like to enjoy specific kittens longer than 120 days and still being able to breed them?
I can imagine this if you are really attached to a kitty you don't want to feel hurried to breed or lose breeding days.
Breeders often have a plan as for breeding such as to combine certain genes from mom and dad and hope to create a kitty which has much visable traits of both mom and dad combined in the offspring.
When this is the case and for example breeders make an other kitty with the other gender they often combine these two to maker a "better" generation and therefore as for the traits of the initial kitties their offsprings are "better" and this makes their parents expire in a way for the people who purely breed for traits.
Why in most instances we don't even use the full 120 days a kitty could breed:-)
I notice you look more for a long term pet relationship with a certain kitty without the need to breed within the given days but to do this in a more natural way when you feel like making a kitty with them as with real pets.
I like this idea actually because honestly eventhough I breed for traits and to make the trait combinations "better" I sometimes regret we have to detach from a cute kitty as we are mainly focussed on their genes and make them better over having the real pet feeling.
And to answer your last question indeed if you have and enjoyed a kitty for more than 120 days breeding after this is not possible anymore though you can still have it as non breedable pet which eats 50% less than a cat at a breedable age.
Therefore I can imagine you mention you would like to meet the Kitties in an other way in which you don't feel a deadline.
I remember I felt the same with my first kittens when I just started with KittyCatS I just didn't want them to grow old and I absolutely didn't want to menagerie them.
I think we all have experienced this why it would be absolutely chill to have no deadline
