SQL Server 2022

By using db<>fiddle, you agree to license everything you submit by Creative Commons CC0.

CREATE TABLE my_table  
(
    Id	INT,
    Serial_Number	INT,
    Last_update_date	VARCHAR(512),
    Flag	VARCHAR(512)
);

INSERT INTO my_table  (Id, Serial_Number, Last_update_date, Flag) VALUES
	('1', '3344', '07-05-2024', 'N'),
	('2', '3344', '02-05-2024', 'N'),
	('3', '9098', '20-02-2024', 'Y'),
	('4', '9098', '20-02-2024', 'Y'),
	('5', '2323', '07-03-2023', 'Y'),
	('6', '2323', '17-10-2023', 'Y'),
	('7', '2323', '17-10-2023', 'Y'),
	('8', '1894', '10-07-2022', 'N'),
	('9', '1894', '20-02-2024', 'N'),
	('10', '1894', '02-05-2024', 'N');

SELECT * FROM my_table;

SELECT Min(ID) as MinId, [Serial_number], Last_update_date , count(*) as DateCount
FROM my_table
GROUP BY [Serial_number], Last_update_date
HAVING Count(*) > 1 ;

WITH CTE as
(
SELECT Serial_number, count(*) as DateCount
FROM my_table
GROUP BY Serial_number, Last_update_date
HAVING Count(*) > 1  
)
SELECT m.*, 
  CASE WHEN DateCount > 1 THEN 'Y' ELSE 'N' END as CalculatedFlag
FROM my_table m
LEFT JOIN CTE on CTE.Serial_number=m.Serial_number

Id	Serial_Number	Last_update_date	Flag
1	3344	07-05-2024	N
2	3344	02-05-2024	N
3	9098	20-02-2024	Y
4	9098	20-02-2024	Y
5	2323	07-03-2023	Y
6	2323	17-10-2023	Y
7	2323	17-10-2023	Y
8	1894	10-07-2022	N
9	1894	20-02-2024	N
10	1894	02-05-2024	N

MinId	Serial_number	Last_update_date	DateCount
6	2323	17-10-2023	2
3	9098	20-02-2024	2

Id	Serial_Number	Last_update_date	Flag	CalculatedFlag
1	3344	07-05-2024	N	N
2	3344	02-05-2024	N	N
3	9098	20-02-2024	Y	Y
4	9098	20-02-2024	Y	Y
5	2323	07-03-2023	Y	Y
6	2323	17-10-2023	Y	Y
7	2323	17-10-2023	Y	Y
8	1894	10-07-2022	N	N
9	1894	20-02-2024	N	N
10	1894	02-05-2024	N	N