Dict.nb

SetAttributes[Dictionary, Orderless];
Dictionary[key_ -> val_, ___][key_] := val;
Dictionary[___][key_] := key;
d_Dictionary.key_ ^:= d[key];
x_ /. Dictionary[rs___] ^:= x /. {rs};
Dictionary[{rs___}] := Dictionary[rs];

Assoc[Dictionary[key_ -> _, rs___], key_ -> val_] := 
  Dictionary[key -> val, rs];
Assoc[Dictionary[rs___], key_ -> val_] := Dictionary[key -> val, rs];
Assoc[map_Dictionary, rs : (_ -> _) ..] := Fold[Assoc, map, {rs}];
map_Dictionary /. ((Dictionary | List)[rs___] | r_) ^:= 
  Assoc[map, r, rs];
  
Dissoc[Dictionary[key_ -> _, rs___], key_] := Dictionary[rs];
Dissoc[map_Dictionary, keys__] := Fold[Dissoc, map, {keys}];
Dissoc[map_Dictionary, key_] := map;

Example.nb

m = Dictionary[a->1, b->2]

m.a + m.b == m[a] + m[b] == (a + b) /. m == 3

m.c == m[c] == c /. m == c

Assoc[m, b -> 4, c -> 5] == m /. {b -> 4, d -> 5} == Dictionary[a -> 1, b -> 4, c -> 5]

Dissoc[m, a] == Dictionary[b->2]

Minimal.nb

SetAttributes[Dictionary, Orderless];
Dictionary[key_ -> val_, ___][key_] := val;
Dictionary[___][key_] := key;

Assoc[Dictionary[key_ -> _, rs___], key_ -> val_] := Dictionary[key -> val, rs];
Assoc[Dictionary[rs___], key_ -> val_] := Dictionary[key -> val, rs];

Dissoc[Dictionary[key_ -> _, rs___], key_] := Dictionary[rs];
Dissoc[map_Dictionary, key_] := map;