Seed if sown directly taken from fruit don't need stratification but if longer time are out of fruit then i better to stratify them in fridge 3 month.
Young seedlings is best one or two years grow in pot and than plant outside.
When you grow from seed kaki you will not know when tree will start to bloom and how will be quality of fruits, often from seed grown tree need more than 5 years to start blooming first time.
Grafted plants can start blooming sometimes in second year after planting.
In this video you can see difference between diospyros lotus and diospyros kaki grafted plants.
I've only tried a few germination experiments on seeds from store fruit because current cultivars often have very few seeds. So, not enough data to say.
However I have germinated a great many American persimmon (Diospyros virginiana) seeds and I find they respond well to winter sowing: just tuck them in the ground or into a pot once the ground temp has cooled down (you don't want immediate germination, though I'm not sure if it's possible or a problem) and let them weather in place. They come up well in the spring, but quite late (like, in June) after you have given up on all your winter-sowing experiments and planted other things in the same places/pots. Ask me how I know, LOL...