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A combination of OUTER APPLY and ROW_NUMBER seems to do the trick:

WITH CTE AS
(
    SELECT  *,
            Tot_Cnt = ROW_NUMBER() OVER(ORDER BY dt)
    FROM #x
)
SELECT  A.dt,
        A.[Name],
        B.Unq_Cnt,
        A.Tot_Cnt,
        CONVERT(NUMERIC(10,4),B.Unq_Cnt)/A.Tot_Cnt Ratio_of_Cnts
FROM CTE A
OUTER APPLY (SELECT COUNT(DISTINCT [Name]) Unq_Cnt
             FROM CTE
             WHERE dt <= A.dt) B
;

Here is a demo of this.

The last 2 rows of your sample seems to be getting wrong results though, since it should be 5 Unq_Cnt