WITH
  last_year
AS
(
  SELECT
    plate,
    COUNT(*)        AS freq
  FROM
    POS.dbo.plate
  WHERE
    datetime >= CAST(DATEADD(MONTH, -1, DATEFROMPARTS(YEAR(CURRENT_TIMESTAMP) - 1, MONTH(CURRENT_TIMESTAMP), 1)) AS DATE)
  GROUP BY
    plate
  HAVING
    MAX(datetime) >= DATEFROMPARTS(YEAR(CURRENT_TIMESTAMP), MONTH(CURRENT_TIMESTAMP), 1)
)
SELECT
  SUM(CASE WHEN freq > 1 THEN 1.0 ELSE 0.0 END)
  /
  COUNT(*)
    AS percent_returnees
FROM
  last_year

Only look at plates that have been here in the last year.

Count their frequency of visits in that year.

Keep only those that visited in the last month.

Divide "visited more than once in the year" by "all"

The maths in your expected results appears to be wrong, or your sample data is wrong; I don't know which. You state that you have 17 plates in your sample data, 13 that appear once, and 4 that appear more than once, but there are only 17 rows in your sample data. If you you 13 rows that have a distinct value once that only leaves 4 rows, which means that if they are duplicates there are only be 2 different plates, not 4.

Looking at your sample data, 4 plates do appear twice, however, only 9 appear once; I don't know where you got the other 4 from. If we use those numbers, you get ~69% and ~31%, not 76% and 24%.

WITH Visits AS(
    SELECT COUNT(*) AS VisitNumber
    FROM(VALUES(15736,'TUT_UK_70-DEU','2023-02-09T14:09:59.0000000','2023-02-21T12:50:19.400'),
               (15737,'TUT_MW_941-DEU','2023-02-09T14:11:09.0000000','2023-02-21T12:50:19.403'),
               (15738,'TUT_RP_88-DEU','2023-02-09T14:13:11.0000000','2023-02-21T12:50:19.417'),
               (15739,'TUT_KB_666-DEU','2023-02-09T14:14:06.0000000','2023-02-21T12:50:19.417'),
               (15740,'TUT_IP_22-DEU','2023-02-09T14:16:23.0000000','2023-02-21T12:50:19.417'),
               (15741,'SIG_FR_608-DEU','2023-02-09T14:17:19.0000000','2023-02-21T12:50:19.420'),
               (15742,'TUT_EA_39-DEU','2023-02-09T14:21:00.0000000','2023-02-21T12:50:19.420'),
               (15743,'TUT_F_1976-DEU','2023-02-09T14:23:05.0000000','2023-02-21T12:50:19.420'),
               (15744,'TUT_UH_11-DEU','2023-02-09T14:25:52.0000000','2023-02-21T12:50:19.420'),
               (15745,'TUT_KF_509-DEU','2023-02-09T14:26:41.0000000','2023-02-21T12:50:19.420'),
               (15746,'TUT_BZ_999-DEU','2023-02-09T14:27:30.0000000','2023-02-21T12:50:19.420'),
               (15747,'TUT_M_995-DEU','2023-02-09T14:29:54.0000000','2023-02-21T12:50:19.420'),
               (15748,'TUT_JR_247-DEU','2023-02-09T14:33:38.0000000','2023-02-21T12:50:19.423'),
               (15749,'TUT_KF_509-DEU','2023-02-09T15:26:41.0000000','2023-02-21T12:50:19.420'),
               (15750,'TUT_BZ_999-DEU','2023-02-09T15:27:30.0000000','2023-02-21T12:50:19.420'),
               (15751,'TUT_M_995-DEU','2023-02-09T15:29:54.0000000','2023-02-21T12:50:19.420'),
               (15752,'TUT_JR_247-DEU','2023-02-09T15:33:38.0000000','2023-02-21T12:50:19.423'))V(plate_id,plate,datetime,updated_at)
    WHERE V.datetime >= DATEADD(YEAR, -1, SYSDATETIME())
    GROUP BY Plate)
SELECT AVG(CASE VisitNumber WHEN 1 THEN 1. ELSE 0. END) AS NewCustomers,
       AVG(CASE VisitNumber WHEN 1 THEN 0. ELSE 1. END) AS ExistingCustomers
FROM Visits;

I am guessing that the column [datetime] is the date that the customer came for a car wash so what you could do something like this

SELECT final.result,customers_count * 100.0 /SUM(customers_count)
FROM
(SELECT 'Existing' AS result, existing.customers_count 
FROM
    (SELECT SUM(COUNT(plate)) OVER() AS customers_count
    FROM POS.dbo.plate 
    WHERE DATEDIFF(MONTH,GETDATE(),[datetime])<=12
    GROUP BY plate 
    HAVING COUNT(plate) >1) existing
UNION 
SELECT 'Newcomers' AS result, newcomers.customers_count
FROM
    (SELECT SUM(COUNT(plate)) OVER() AS customers_count
    FROM POS.dbo.plate 
    WHERE DATEDIFF(MONTH,GETDATE(),[datetime])<=12
    GROUP BY plate 
    HAVING COUNT(plate)=1) newcomers)final

What I did is find the number of people that came for more than one time and the number of people that came exactly once and in the end I calculated the percentages.