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Natural weapons are considered to be melee weapons rather than unarmed.
I see in this answer that natural weapons are neither simple nor martial weapons, but this seems to be an assumption that I can't find in any rulebooks or the errata.
All I can find is this relatively out of context quote from the equipment section in the Player's Handbook:

Your race, class, and feats can grant you proficiency
with certain weapons or categories of weapons. The two
categories are simple and martial.

This line implies that every weapon is considered either simple or martial. It could be assumed to only refer to equipment, or only refer to proficiency, but I can't find anything excluding natural weapons from this.

Are natural weapons able to be simple or martial weapons, or am I missing something that puts them into a third exclusive category?

I would like to multiclass as a monk/druid, and choose a natural weapon such as a bear's claws as my Kensei weapon. It sounds like that's not completely within the rules (and I haven't run the numbers, so I can't tell you whether it's effective). I'm just wondering if I missed something here.

D&D 5e lead rules designer Jeremy Crawford has issued a clear clarification on this one:

Natural weapons are neither simple nor martial weapons. They're their own thing.

The rule you quoted does not say that all weapons are either simple or martial, it says that the two categories of weapons you can be proficient in due to your race, class or feats are simple and martial.