SQL Server 2014

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Name	Start Date	End Date
Joe	2021-04-20	2021-05-20
John	2021-05-01	2021-05-28

WITH 
cte1 AS ( SELECT MIN(MONTH([Start Date])) [Month], 
                 MAX(MONTH([End Date])) LastMonth, 
                 MIN(YEAR([Start Date])) [Year]
          FROM test
          UNION ALL
          SELECT [Month] + 1, 
                 LastMonth,
                 [Year]
          FROM cte1
          WHERE [Month] < LastMonth ),
cte2 AS ( SELECT DATEFROMPARTS([Year], [Month], 1) MonthStart,
                 EOMONTH(DATEFROMPARTS([Year], [Month], 1)) MonthEnd,
                 [Month]
          FROM cte1 )
SELECT cte2.[Month],
       SUM(1 + DATEDIFF(day, CASE WHEN test.[Start Date] < cte2.MonthStart
                                  THEN cte2.MonthStart
                                  ELSE test.[Start Date] END,
                             CASE WHEN test.[End Date] > cte2.MonthEnd
                                  THEN cte2.MonthEnd
                                 ELSE test.[End Date] END )) [Count]
FROM test
JOIN cte2 ON test.[Start Date] <= cte2.MonthEnd
         AND cte2.MonthStart <= test.[End Date]
GROUP BY cte2.[Month];

Month	Count
4	11
5	48