I don't think there a conflict between those two meanings of the word, one an adjective and one a verb, but rather that the connection between them can be obscure.
When something is appropriate it is proper: being proper is an attribute or property of the thing. Likewise, when something is deemed proper, that attribution has been established, defined, and in a sense made "real." Here you find that the sense of "real" -- solid, firm, land as in real estate -- and property, something owned, is lurking not too far (ahem) afield from appropriate: something is appropriate when it appears as a valid, acceptable, or real property of some other thing under consideration.
When something is appropriated, its property and ownership status as belonging to or being attributed to (say) person A is shifted or transferred to person B. An appropriated object (etc) contains ownership and property traces of both the former and current "proprietor."
The common link between the two meanings/senses of the word, as adjective and verb, is through the idea of ownership and attribution, the issue being one of a proper belonging-to.
So: a behavior or action is inappropriate when it is seen as not a "proper" attribute -- owned or native characteristic -- of the actor.
A thing is considered appropriated when its possession is seen as not a "proper" attribute -- a native or legitimate characteristic -- of its current proprietor.
The root of the word(s) is in the idea ownership and property.