If you want to count rows by month , use default group by month - this will group all rows which fall in a single month.
To fetch the count of rows in a single month, your query should be

SELECT Royalty_Month, count(Event_Combined) as total FROM agency
group by Royalty_Month, MONTH(Royalty_Month) // full group by
ORDER BY Royalty_Month ASC

which will generate

Live example
Read more on default date time functions here

In MySQL 8.0, one option is to:

• retrieve first time an event appears, using a ranking value given by the ROW_NUMBER window function
• count only first appearances for each day, with a conditional sum of 1s

WITH cte AS (
    SELECT *,
           ROW_NUMBER() OVER(PARTITION BY Event_Combined ORDER BY Royalty_Month) AS rn
    FROM tab
)
SELECT Royalty_Month,
       SUM(IF(rn = 1, 1, 0)) AS Total_Events
FROM cte
GROUP BY Royalty_Month

you can try this on previous versions as well as new version of MySQL:

SELECT  CurrMonth,  SUM(CASE WHEN Event_PrevMonth IS NULL THEN 1 ELSE 0 END)TOTAL_EVENTS
FROM
(
    SELECT DISTINCT A1.Event_Combined, A1.Royalty_Month CurrMonth, A2.Event_Combined Event_PrevMonth , A2.Royalty_Month Prev_month
    FROM Agency A1
    LEFT OUTER JOIN
    (
        SELECT Event_Combined, Royalty_Month 
        FROM Agency
    )A2 ON A1.Event_Combined = A2.Event_Combined AND A1.Royalty_Month > A2.Royalty_Month
) T 
GROUP BY CurrMonth;