dnd 5e – Can natural weapons be simple or martial weapons?

Natural weapons are considered melee weapons instead of unarmed ones.
I see in this answer that natural weapons are neither simple nor martial weapons, but this seems to be an assumption that I can not find in any regulation or errata.
All I can find is this quote relatively out of context of the team section in the Player's Manual:

Your race, class and deeds can give you dominance
With certain weapons or categories of weapons. Both
The categories are simple and martial.

This line implies that each weapon is considered simple or martial. You could assume that it only refers to the team, or only the competition, but I can not find anything that excludes the natural weapons of this.

Can natural weapons be simple or martial weapons, or am I missing something that puts them in an exclusive third category?

I would like to make a multiclass as a monk / druid, and choose a natural weapon like the claws of a bear like my Kensei weapon. It seems that it is not completely within the rules (and I have not run the numbers, so I can not tell you if it's effective). I wonder if I missed something here.