Dragons that owns a domain or realm tend to favor kobolds as their workforce, mostly since kobolds are related to dragonkin, thus easier to deal with for dragons. They tend to adapt to whichever dragon's influence is closest to them, which tend to be their patron dragon. So over time, especially after a couple generations, a clan of kobold will have a similar look/color/traits as the dragon patron.

Tho some dragons do have other species working the same, but none have the same clan mentality or ease of adaptation. So I'd see those forming a worshipping cult instead of treating the dragon as a kind of political authority or reigning noble.

In many other cases, dragons don't have a kobold clan under them, be it because they don't care to own land, or don't have enough influence (or only have a tiny group). Similarly, many kobold clans go without dragons, be it by choice, or simply they haven't found -THE- dragon they feel destined to join with. There might be other strong enough creatures a kobold clan might pick (they can be like cats), but most other species find them too hard to wrangle. They're not 100% bottoms, only 92,3%.