Your two indices do exactly the same thing, so you can remove the second one, it's useless.
To optimize your query, the order of the fields inside the index must be reversed:
create index report_lp_datetime_index on report (datetime,license_plate_id);
BEGIN;
CREATE TABLE foo (d INTEGER, i INTEGER);
INSERT INTO foo SELECT random()*100000, random()*1000 FROM generate_series(1,1000000) s;
CREATE INDEX foo_d_i ON foo(d DESC,i);
COMMIT;
VACUUM ANALYZE foo;
EXPLAIN ANALYZE SELECT * FROM foo WHERE i IN (1,2,4,5,6,7,8,10,15,22,34,75) ORDER BY d DESC LIMIT 100;
Limit (cost=0.42..343.92 rows=100 width=8) (actual time=0.076..9.359 rows=100 loops=1)
-> Index Only Scan Backward using foo_d_i on foo (cost=0.42..40976.43 rows=11929 width=8) (actual time=0.075..9.339 rows=100 loops=1)
Filter: (i = ANY ('{1,2,4,5,6,7,8,10,15,22,34,75}'::integer[]))
Rows Removed by Filter: 9016
Heap Fetches: 0
Planning Time: 0.339 ms
Execution Time: 9.387 ms
Note the index is not used to optimize the WHERE clause. It is used here as a compact and fast way to store references to the rows ordered by date DESC, so the ORDER BY can do an index-only scan and avoid sorting. By adding column id to the index, an index-only scan can be performed to test the condition on id, without hitting the table for every row. Since there is a low LIMIT value it does not need to scan the whole index, it only scans it in date DESC order until it finds enough rows satisfying the WHERE condition to return the result.
It will be faster if you create the index in date DESC order, this could be useful if you use ORDER BY date DESC + LIMIT in other queries too.
You forget that OP's table has a third column, and he is using SELECT *. So that wouldn't be an index-only scan.
Easy to work around. The optimum way to do this query would be an index-only scan to filter on WHERE conditions, then LIMIT, then hit the table to get the rows. For some reason if "select *" is used postgres takes the id column from the table instead of taking it from the index, which results in lots of unnecessary heap fetches for rows whose id is rejected by the WHERE condition.
Easy to work around, by doing it manually. I've also added another bogus column to make sure the SELECT * hits the table.
EXPLAIN (ANALYZE,buffers) SELECT * FROM foo
JOIN (SELECT d,i FROM foo WHERE i IN (1,2,4,5,6,7,8,10,15,22,34,75) ORDER BY d DESC LIMIT 100) f USING (d,i)
ORDER BY d DESC LIMIT 100;
Limit (cost=0.85..1281.94 rows=1 width=17) (actual time=0.052..3.618 rows=100 loops=1)
Buffers: shared hit=453
-> Nested Loop (cost=0.85..1281.94 rows=1 width=17) (actual time=0.050..3.594 rows=100 loops=1)
Buffers: shared hit=453
-> Limit (cost=0.42..435.44 rows=100 width=8) (actual time=0.037..2.953 rows=100 loops=1)
Buffers: shared hit=53
-> Index Only Scan using foo_d_i on foo foo_1 (cost=0.42..51936.43 rows=11939 width=8) (actual time=0.037..2.935 rows=100 loops=1)
Filter: (i = ANY ('{1,2,4,5,6,7,8,10,15,22,34,75}'::integer[]))
Rows Removed by Filter: 9010
Heap Fetches: 0
Buffers: shared hit=53
-> Index Scan using foo_d_i on foo (cost=0.42..8.45 rows=1 width=17) (actual time=0.005..0.005 rows=1 loops=100)
Index Cond: ((d = foo_1.d) AND (i = foo_1.i))
Buffers: shared hit=400
Execution Time: 3.663 ms
Another option is to just add the primary key to the date,license_plate index.
SELECT * FROM foo JOIN (SELECT id FROM foo WHERE i IN (1,2,4,5,6,7,8,10,15,22,34,75) ORDER BY d DESC LIMIT 100) f USING (id) ORDER BY d DESC LIMIT 100;
Limit (cost=1357.98..1358.23 rows=100 width=17) (actual time=3.920..3.947 rows=100 loops=1)
Buffers: shared hit=473
-> Sort (cost=1357.98..1358.23 rows=100 width=17) (actual time=3.919..3.931 rows=100 loops=1)
Sort Key: foo.d DESC
Sort Method: quicksort Memory: 32kB
Buffers: shared hit=473
-> Nested Loop (cost=0.85..1354.66 rows=100 width=17) (actual time=0.055..3.858 rows=100 loops=1)
Buffers: shared hit=473
-> Limit (cost=0.42..509.41 rows=100 width=8) (actual time=0.039..3.116 rows=100 loops=1)
Buffers: shared hit=73
-> Index Only Scan using foo_d_i_id on foo foo_1 (cost=0.42..60768.43 rows=11939 width=8) (actual time=0.039..3.093 rows=100 loops=1)
Filter: (i = ANY ('{1,2,4,5,6,7,8,10,15,22,34,75}'::integer[]))
Rows Removed by Filter: 9010
Heap Fetches: 0
Buffers: shared hit=73
-> Index Scan using foo_pkey on foo (cost=0.42..8.44 rows=1 width=17) (actual time=0.006..0.006 rows=1 loops=100)
Index Cond: (id = foo_1.id)
Buffers: shared hit=400
Execution Time: 3.972 ms
Edit
After thinking about it... since the LIMIT restricts the output to 100 rows ordered by date desc, wouldn't it be nice if we could get the 100 most recent rows for each license_plate_id, put all that into a top-n sort, and only keep the best 100 for all license_plate_ids? That would avoid reading and throwing away a lot of rows from the index. Even if that's much faster than hitting the table, it will still load up these index pages in RAM and clog up your buffers with stuff you don't actually need to keep in cache. Let's use LATERAL JOIN:
EXPLAIN (ANALYZE,BUFFERS)
SELECT * FROM foo
JOIN (SELECT d,i FROM
(VALUES (1),(2),(4),(5),(6),(7),(8),(10),(15),(22),(34),(75)) idlist
CROSS JOIN LATERAL
(SELECT d,i FROM foo WHERE i=idlist.column1 ORDER BY d DESC LIMIT 100) f2
ORDER BY d DESC LIMIT 100
) f3 USING (d,i)
ORDER BY d DESC LIMIT 100;
It's even faster: 2ms, and it uses the index on (license_plate_id,date) instead of the other way around. Also, and this is important, since each subquery in the lateral hits only the index pages that contain rows that will actually be selected, while the previous queries hit much more index pages. So you save on RAM buffers.
If you don't need the index on (date,license_plate_id) and don't want to keep a useless index, that could be interesting since this query doesn't use it. On the other hand, if you need the index on (date,license_plate_id) for something else and want to keep it, then... maybe not.
Please post results for the winning query