We need a unique column for able to make the comparison at the update operation after the insert. That's why we are using ExternalID column temporarily. ExternalID updated by row_nubmber.
;WITH CTE AS
(
    SELECT *, RN = ROW_NUMBER() OVER(ORDER BY (SELECT NULL)) FROM @TempTable
)
UPDATE CTE SET ExternalID = RN
We are keeping  the output of the insert operation in a temp table. The trick is order by with ExternalID, it will help us for making the unique row number for same first and last name
DECLARE @output TABLE (
ID  INT,
FirstName VARCHAR(10),
LastName VARCHAR(10))
Insert into @myTable
OUTPUT inserted.ID, inserted.FirstName, inserted.LastName  INTO @output(ID, FirstName, LastName) 
select FirstName, LastName from @TempTable T 
order by ExternalID
For replacing the ExternalID column with inserted id value, we are making comparing with first name, last name and row number.
;WITH TMP_T AS(
    SELECT *, RN = ROW_NUMBER() OVER(PARTITION BY FirstName, LastName ORDER BY ExternalID) FROM @TempTable )
,OUT_T AS( 
    SELECT *, RN = ROW_NUMBER() OVER(PARTITION BY FirstName, LastName ORDER BY ID) FROM @output )
UPDATE TMP_T SET ExternalID = OUT_T.ID
FROM 
TMP_T INNER JOIN OUT_T ON 
    TMP_T.FirstName = OUT_T.FirstName 
    AND TMP_T.LastName = OUT_T.LastName 
    AND TMP_T.RN = OUT_T.RN
Sample Data:
DECLARE @TempTable TABLE (
FirstName VARCHAR(10),
LastName VARCHAR(10),
DOB VARCHAR(10),
Sex VARCHAR (10),
Age VARCHAR(10),
ExternalID INT)
INSERT INTO @TempTable VALUES
('Serkan1', 'Arslan1', 'A','M','1',NULL),
('Serkan2', 'Arslan2', 'B','M','1',NULL),
('Serkan3', 'Arslan', 'C','M','1',NULL),
('Serkan3', 'Arslan', 'D','M','1',NULL)
DECLARE @myTable TABLE (
ID  INT identity(100,1), -- started from 100 for see the difference
FirstName VARCHAR(10),
LastName VARCHAR(10))
Result:
MyTable
ID          FirstName  LastName
----------- ---------- ----------
100         Serkan1    Arslan1
101         Serkan2    Arslan2
102         Serkan3    Arslan
103         Serkan3    Arslan
TempTable
FirstName  LastName   DOB        Sex        Age        ExternalID
---------- ---------- ---------- ---------- ---------- -----------
Serkan1    Arslan1    A          M          1          100
Serkan2    Arslan2    B          M          1          101
Serkan3    Arslan     C          M          1          102
Serkan3    Arslan     D          M          1          103