SQL Server 2022

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-- data prep part
SELECT *
INTO #t
FROM (    --   DROP TABLE IF EXISTS  #t
          SELECT 100 ID,  CAST('2022-03-02' AS DATE) eStart ,  CAST('2022-03-15' AS DATE) eEnd, '1 Same Month island' note , 100 Idx
    UNION SELECT 200,  '2022-01-11' ,   '2022-03-27', '2 Same Month Start 1', 2001
    UNION SELECT 200,  '2022-02-15' ,   '2022-5-15', '2 Same Month Start 2', 2002
    UNION SELECT 300,  '2021-05-08' ,   '2024-12-31', '3 Future eEnd ' ,3000
    UNION SELECT 440,  '2022-10-15' ,   '2023-10-10', '4 Future end 2' ,4400 
) b

SELECT * FROM #t

SELECT
    A.MonthStart,
    EOMONTH(A.MonthStart) AS MonthEnd
INTO #months
FROM (VALUES (1), (2), (3), (4), (5), (6), (7), (8), (9), (10), (11), (12)) N(N)
CROSS APPLY (SELECT DATEFROMPARTS(2022, N.N, 1) AS MonthStart) A

SELECT * from #months

SELECT
    t.ID,
    CASE WHEN m.MonthStart >= t.eStart THEN m.MonthStart ELSE t.eStart END AS eStart,
    CASE WHEN m.MonthEnd <= t.eEnd THEN m.MonthEnd ELSE t.eEnd END AS eEnd,
    t.Idx
FROM #t t
JOIN #months m -- Overlapping months (inclusive dates)
    ON m.MonthStart <= t.eEnd 
    AND t.eStart <= m.MonthEnd 
ORDER BY t.ID, t.Idx, eStart -- Added t.Idx to better atch desired results

ID	eStart	eEnd	note	Idx
100	2022-03-02	2022-03-15	1 Same Month island	100
200	2022-01-11	2022-03-27	2 Same Month Start 1	2001
200	2022-02-15	2022-05-15	2 Same Month Start 2	2002
300	2021-05-08	2024-12-31	3 Future eEnd	3000
440	2022-10-15	2023-10-10	4 Future end 2	4400

MonthStart	MonthEnd
2022-01-01	2022-01-31
2022-02-01	2022-02-28
2022-03-01	2022-03-31
2022-04-01	2022-04-30
2022-05-01	2022-05-31
2022-06-01	2022-06-30
2022-07-01	2022-07-31
2022-08-01	2022-08-31
2022-09-01	2022-09-30
2022-10-01	2022-10-31
2022-11-01	2022-11-30
2022-12-01	2022-12-31

ID	eStart	eEnd	Idx
100	2022-03-02	2022-03-15	100
200	2022-01-11	2022-01-31	2001
200	2022-02-01	2022-02-28	2001
200	2022-03-01	2022-03-27	2001
200	2022-02-15	2022-02-28	2002
200	2022-03-01	2022-03-31	2002
200	2022-04-01	2022-04-30	2002
200	2022-05-01	2022-05-15	2002
300	2022-01-01	2022-01-31	3000
300	2022-02-01	2022-02-28	3000
300	2022-03-01	2022-03-31	3000
300	2022-04-01	2022-04-30	3000
300	2022-05-01	2022-05-31	3000
300	2022-06-01	2022-06-30	3000
300	2022-07-01	2022-07-31	3000
300	2022-08-01	2022-08-31	3000
300	2022-09-01	2022-09-30	3000
300	2022-10-01	2022-10-31	3000
300	2022-11-01	2022-11-30	3000
300	2022-12-01	2022-12-31	3000
440	2022-10-15	2022-10-31	4400
440	2022-11-01	2022-11-30	4400
440	2022-12-01	2022-12-31	4400