There is another pair of shapes. I did a complete search but only to size 10 for one piece. The solution above has pieces with area 4 and 10, the second I found has 7 and 7. I show them both. Note that there are four distinct positions for the domino hole, all others are rotations/reflections of these.
For completeness, I show the 14 ways of doing this with three pieces (with a maximum size of 10 for any piece). Note that four of them have two congruent pieces. Three congruent is not possible.
If you change the missing area to a single cell instead, nothing much changes for the two-piece case.
If you allow pieces as small as three and as large as 10 you get a bunch of extra cases. Here are the three piece tilings. Unfortunately none exists with three copies of one pentomino, although there are a number with three pentominoes, some of them with two copies of one pentomino.
And of course the four- and five-piece tilings for the missing monomino case:
For a 5x5: 3 piece missing domino and monomino tilings, searched only pieces of size 3 to 12
For the 5x5 missing domino case there are also 1258 4-piece tilings, 5492 5-piece tilings, 2179 with 6 pieces and 89 with 7 pieces. 'Nicest' 7-piece case shown with 5 of the L-tromino and 2 of the I-tetromino
Counts for the 5x5 missing domino case were: 4 pieces: 1064. 5 pieces:10847. 6 pieces: 6822. 7 pieces: 388
And of course some 8-piece tilings since we have the possibility of 8 trominoes with area 24. Just four possibilities: