You can use distinct in the subquery used for the apply:

SELECT d.ProjectID,
       STRING_AGG(  newID.value, ',') WITHIN GROUP (ORDER BY newID.value) AS 
NewField
FROM [dbo].[Data] d CROSS APPLY
     (select distinct value
      from STRING_SPLIT(d.[bID], ';') AS newID 
     ) newID
WHERE newID.value IN (   'O95833' , 'Q96NY7-2'  ) 
group by projectid;

As @SeanLange pointed out in the comments, this is a terrible way to pull out the data, but if you had to, just make it 2 separate queries as follows:

SELECT 
    ProjectID
    ,STRING_AGG( val, ',') WITHIN GROUP (ORDER BY val) AS NewField
FROM
(
    SELECT DISTINCT 
        ProjectID
        ,newID.value AS val
    FROM 
        [dbo].[Data] WITH(NOLOCK)  
        CROSS APPLY STRING_SPLIT([bID],';') AS newID  
    WHERE 
        newID.value IN ('O95833' , 'Q96NY7-2') 
) t
GROUP BY
    ProjectID

That should do it.

This is a function that I wrote that answers the OP Title: Improvements welcome!

CREATE OR ALTER FUNCTION [dbo].[fn_DistinctWords]
(
  @String NVARCHAR(MAX)  
)
RETURNS NVARCHAR(MAX)
WITH SCHEMABINDING
AS
BEGIN
  DECLARE @Result NVARCHAR(MAX);
  WITH MY_CTE AS ( SELECT Distinct(value) FROM STRING_SPLIT(@String, ' ')  )
  SELECT @Result = STRING_AGG(value, ' ') FROM MY_CTE
  RETURN @Result
END
GO

Use like:

SELECT dbo.fn_DistinctWords('One Two      Three Two One');

You can make a distinct view of the table, that holds the aggregate values, that is even simpler:

Create Table Test (field1 varchar(1), field2 varchar(1));

go

Create View DistinctTest as (Select distinct field1, field2 from test group by field1,field2);

go

insert into Test Select 'A', '1';
insert into Test Select 'A', '2';
insert into Test Select 'A', '2';
insert into Test Select 'A', '2';
insert into Test Select 'D', '1';
insert into Test Select 'D', '1';

select string_agg(field1, ',')  from Test where field2 = '1';  /* duplicates: A,D,D */;

select string_agg(field1, ',')  from DistinctTest where field2 = '1';  /* no duplicates: A,D  */;

Another possibility to get unique strings from STRING_AGG would be to perform these three steps after fetching the comma separated string:

1. Split the string (STRING_SPLIT)
2. Select DISTINCT from the splits
3. Apply STRING_AGG again to a select with a group on a single key

Example:

(select STRING_AGG(CAST(value as VARCHAR(MAX)), ',') 
        from (SELECT distinct 1 single_key, value 
            FROM STRING_SPLIT(STRING_AGG(CAST(customer_division as VARCHAR(MAX)), ','), ',')) 
                q group by single_key) as customer_division

Here is my improvement on @ttugates to make it more generic:

CREATE OR ALTER FUNCTION [dbo].[fn_DistinctList]
(
  @String NVARCHAR(MAX),
  @Delimiter char(1)
)
RETURNS NVARCHAR(MAX)
WITH SCHEMABINDING
AS
BEGIN
  DECLARE @Result NVARCHAR(MAX);
  WITH MY_CTE AS ( SELECT Distinct(value) FROM STRING_SPLIT(@String, 
@Delimiter)  )
  SELECT @Result = STRING_AGG(value, @Delimiter) FROM MY_CTE
  RETURN @Result
END

Oracle (since version 19c) suports listagg (DISTINCT ..., but Microsoft SQL Server not probably.