extracting sublists

SequenceCases[lis, {DateObject[{2019, 1, 2}], a:Except[_DateObject] ..., _DateObject} :> a]

{"c", "d", "e"}

Also

Cases[lis,
  {___, DateObject[{2019, 1, 2}], a : Except[_DateObject] ..., _DateObject, ___} :> a, All]

{"c", "d", "e"}

f[data_List, d_DateObject] := Append[data, d -> {}]
f[{dates___, d_ -> l_List}, s_String] := {dates, d -> Append[l, s]}
lookup = Fold[f, {}, lis]

We can now use this lookup like this:

{"c", "d", "e"}

lis = {
   DateObject[{2019, 1, 1}], "a", "b",
   DateObject[{2019, 1, 2}], "c", "d", "e",
   DateObject[{2019, 1, 3}], "f"};

f[x_] := Rest[FirstCase[Split[lis, Head[#2] =!= DateObject &], {x, ___}]]

res = f[DateObject[{2019, 1, 2}]]

{"c", "d", "e"}