Is there a unique characteristic for all canonical books? Is there any one unique characteristic that would be common for all the books of the Bible besides being chosen by the Church to be a part of canon? I am interested here in the main-stream Protestant tradition.
Technically, the only "criteria" for canonicity is widespread acceptance.  Looking at canonical lists up through the Festal Letter canonicity was always based on "what seemed most profitable to the reader" at the time. That said, the Council of Trent said


All New Testament Books share a claim to apostolic authority:


Matthew &amp; John were disciples
Luke (&amp;hence Acts) travelled with Paul
Mark was probably a close friend of Peter's
All of the Pauline Epistles (Rom - Philemon) were written by Paul, who is considered an aposotle
Hebrews was mistakenly attributed to Paul
James &amp; Jude were brothers of Jesus
Peter (I &amp; II) &amp; John (I, II, III, and Revelation) were disciples 

The Old Testament books had their authority by the Jews, and several theories including the Council of Jamnia have been put forward that describe their assembly.




By contrast, the NT Apocraphya (Acts of Thecla, Gospel of Barnabas, Gospel of Peter, etc...) were never widely held to have been of apostolic origin, and the Church Fathers (the Didache, I Clement, Matrydom of Polycarp, etc...) were written by people who were more than a generation removed from Jesus.