Japt v2.0a0, 15 13 bytes

My very first "proper" quine. At long fecking last!

oQ+"oQ+ â"Q â

Try it

oQ+"..."Q â     :Implicit input of string
o               :Keep only those characters that appear in
 Q              :  Quotation mark
  +"..."        :  Append literal string
        Q       :  Passed as a second argument to the o method to disable the case insensitive flag (any truthy value would work)
          â     :Deduplicate