Here is how you find the "first" and "last" day of the month (assuming these are the first and last referenced days since day 1 would always be the first day but isn't in your example data.)

WITH numbered AS (
  SELECT *,
      ROW_NUMBER() OVER (PARTITION BY Customer, YEAR(date), MONTH(date) ORDER BY DAY(date) ASC as RN_FIRST,
      ROW_NUMBER() OVER (PARTITION BY Customer, YEAR(date), MONTH(date) ORDER BY DAY(date) DESC as RN_LAST
  FROM sometableyoudidntname
)
SELECT *, F.Date as FIRST_DATE, L.Date as LAST_DATE
FROM sometableyoudidntname X
LEFT JOIN numbered F ON F.Customer = X.Customer AND F.date = X.date AND RN_FIRST = 1
LEFT JOIN numbered L ON L.Customer = X.Customer AND L.date = X.date AND RN_LAST = 1

Now that you know the date of the first and the last items you should be able to do the group by

NOTE: It is not exactly clear to me if I am using the right partition and joins. This depends on what drives the uniqueness of your data model.

It's really helpful if you provide DDL/DML that's easy to reproduce when you ask questions like this. Your result set doesn't seem to match the example data you provided.

Consider:

DECLARE @Table TABLE (Date DATE, Country NVARCHAR(50), Customer_type INT);
INSERT INTO @Table (Date, Country, Customer_type) VALUES 
('2023-03-01', 'Germany', 1),
('2023-03-01', 'Germany', 1),
('2023-03-02', 'France', 3),
('2023-03-05', 'UK', 1),
('2023-03-10', 'Spain', 5),
('2023-03-31', 'Germany', 2);

Using this we just need to be able to flag the first and last day values, and then aggregate on that:

SELECT RIGHT('0'+CAST(DATEPART(MONTH,Date) AS VARCHAR),2)+'/'+RIGHT(CAST(DATEPART(YEAR,Date) AS VARCHAR),2) AS Date, Country, Customer_type, COUNT(a.Opening_Value) AS Opening_Value, AVG(.0+a.Average_Value) AS Average_Value, COUNT(a.Closing_Value) AS Closing_Value
  FROM (
SELECT Date, Country, Customer_type, 
       CASE WHEN DATEPART(DAY,Date) = 1 THEN 1 END AS Opening_Value, 
       1 AS Average_Value, 
       CASE WHEN DATEPART(DAY,DATEADD(DAY,1,Date)) = 1 THEN 1 END AS Closing_Value
  FROM @Table
       ) a
 GROUP BY DATEPART(MONTH,Date), DATEPART(YEAR,Date), Country, Customer_type

As a side note, when using dates it's a good idea to use the ISO format of yyyy-mm-dd as it removes any ambiguity.