It might need some extra tweaking, but see this:

Tweaking like

• Replacing CHAR(13) and/or CHAR(10) but a space before doing anything else.
• A situation like only 1 " or more then 2 " might need more testing.

SELECT
   FirstName,
   SUBSTRING(FirstNameORG,LEN(FirstName)+1,LEN(FirstNameORG)) as MiddleNames
FROM (
   SELECT
      CASE WHEN firstSpace < FirstDoubleQuoot
           THEN SUBSTRING(FirstName,1,firstSpace-1)
           WHEN secondDoublequoot > firstDoublequoot
           THEN SUBSTRING(FirstName,firstDoublequoot-1,secondDoublequoot-firstDoublequoot+2)
           WHEN firstSpace>1
           THEN SUBSTRING(FirstName,1,firstSpace-1)
           ELSE FirstName
      END as FirstName,
     FirstName as FirstNameORG
   FROM (
      SELECT
         FirstName,
         CHARINDEX(' ',FirstName) firstSpace,
         CHARINDEX('"',FirstName) firstDoubleQuoot,
         CHARINDEX('"',FirstName,CHARINDEX('"',FirstName)+2) secondDoubleQuoot
      FROM #Names
   ) x
) y;

output:

see: DBFIDDLE

I got to the solution thanks to @Luuk

• I fogot i only needed to group up the "-Groups when the field starts with them
• I was afraid i had to sanatize the FirstName-Field in each and every parsing, But the subquery solved that one for me.

Here is the Script I'm adapting for my productive solution now:

select ID ADR_ID, First_Name, trim(RIGHT(ORGFirstName, LEN(ORGFirstName) - len(First_Name))) as Middle_Name from
(
    SELECT
        Qua.ID ID,
        Qua.FirstName ORGFirstName,
        case
            when (FstQuote = 1) AND (SndQuote <> 0) then
                LEFT(Qua.FirstName, SndQuote)
            when FstBlank <> 0 then
                LEFT(Qua.FirstName, FstBlank - 1)
            else
                Qua.FirstName
        end as First_Name
    FROM (
        select
            ID,
            FirstName,
            CHARINDEX('"', San.FirstName) FstQuote,
            CHARINDEX('"', San.FirstName, 2) SndQuote,
            CHARINDEX(' ', San.FirstName) FstBlank
        from (
            SELECT
                ID,
                TRIM(REPLACE(REPLACE(REPLACE(REPLACE(FirstName, char(10), ''), char(13), ''), char(9), ' '), '  ', ' ')) FirstName
            FROM #Names) as San --Sanatized
        ) as Qua -- Quantified
) as FirstNamesParsed

There is a limit on how much you can save the users from faulty data. So i decided to ignore lonely quotes and if a user decides to copy-paste 15 tabstops into between two middle names, it will show as well.