I went with a simpler solution:

Update the SELECT DateData to SELECT EOMONTH(DateData) and that solved the issue.

Thank you everyone for the help.

You can use the EOMONTH function. Note that you should still increment off the first day in the month, otherwise you can run into issues.

DECLARE @StartDateTime DATETIME
DECLARE @EndDateTime DATETIME
 
SET @StartDateTime = '2015-01-01'
SET @EndDateTime = '2016-01-12';
 
WITH DateRange AS 
(
    SELECT EOMONTH(@StartDateTime) as DateData, @StartDateTime AS StartOfMonth
    UNION ALL
    SELECT EOMONOTH(DATEADD(m, 1, StartOfMonth))
    FROM DateRange 
    WHERE DateData < @EndDateTime
)
SELECT DateData
FROM DateRange
OPTION (MAXRECURSION 0)

Recursion is slow. GENERATE_SERIES or some other numbers generator is probably better. EOMONTH also provides an increment parameter, which means you don't need DATEADD

DECLARE @StartDateTime DATETIME
DECLARE @EndDateTime DATETIME
 
SET @StartDateTime = '2015-01-01'
SET @EndDateTime = '2016-01-12';
 
SELECT
  EOMONTH(@StartDateTime, g.value) AS DateData
FROM GENERATE_SERIES(0, DATEDIFF(month, @StartDateTime, @EndDateTime)) g;

The format should not be relevant, that is purely a display/presentation thing. But if you really needed to stringify the date, you can use CONVERT(varchar(10), yourDate, 110) .