Is the last of these results expected? If so, how?

assoc=<|"a"->1,"b"->2|>;
assoc["a"]+=1;
assoc (* <|"a" -> 2, "b" -> 2|> *)
assoc[[{"a","b"}]]=1;
assoc  (* <|"a" -> 1, "b" -> 1|> *)
assoc[[{"a","b"}]]+=1;
assoc  (* <|"a" -> <|"a" -> 2, "b" -> 2|>, "b" -> <|"a" -> 2, "b" -> 2|>|> *)

Edit:

Kuba helpfully points out that this reflects the behavior of Part, specifically

assoc = <|"a" -> 1, "b" -> 2|>;
assoc[[{"a", "b"}]]  (* <|"a" -> 1, "b" -> 2|> *)

Unfortunately, this pushes my question back to the behavior of Part. We are told by the docs that Part[expr,i] gives the ith part of expr and that Part[expr,{i1,i2,…}] gives a list of the parts i1, i2, … of expr. This seems to imply that

Part[expr, {i1,i2,...}]==(Part[expr,#]&/@{i1,i2,...})

But this is clearly not the case for associations. E.g.,

assoc[[#]] & /@ {"a", "b"}     (* {1,2} *)

How am I misreading the docs?