There are no "flavours" for each colour, however there are three different tiers of Poké Beans.
1: Regular Poké Beans
These are the beans that are on solid colour. They are the most common by far, and raise affection by a small amount.
2: Uncommon Poké Beans
I don't have a better name for them. These beans are coloured specks on them and are noticeably different from regular Poké Beans. They raise affection by more than regular beans, but are less common.
3: Rainbow Poké Beans
These Poké Beans have a rainbow pattern. They are the rarest Poké Beans. Feeding them to your Pokémon will raise affection by a large amount.
Source: Knowledge, I have fed many Poké Beans to my Pokémon.