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
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.
“He said to him, ‘If they do not listen to Moses and the Prophets, they will not be convinced even if someone rises from the dead.’” Luke 16:31
The key Christian idea is that scripture it is perfect and self-evincing in and of themselves without requiring any approval or acceptance from human authority. From this standpoint it is the ‘internal perfection of scripture’ that places it above all other texts. Any other text a man can easily prove some foul idea, or some imperfection, is contained in them, or some doctrinal error as compared to scripture. So in theory without any church council who previously determined what scripture cannon was; the scripture carries its own full authority and self-evincing nature. The result of this power in the scripture is that any group of sound believers would commonly choose the same books of the Bible based on the Spirits own internal and absolutely persuasive voice emanating from them as did previous believers under previous generations. I say ‘in general’ because not always has every book in our Bible been considered as scripture by all people. However, over time a general consensus of their internal perfection is made that proves those books which can’t be overthrown by man’s opinion.
However in addition to the neat and basic true answer, I would like to add lesser ‘external charcteristsics’ that separate the scripture from other texts in other ways. These charcteristsics do not prove what the scriptures are (only their internal perfection does that) but they serve as 'additional points of interest' to understand how they are unique since they are perfect:
-1- They are very, very old. This may seem funny but it is believed that God would not leave man without any truth so scripture must start as far as history recalls, otherwise we could say, ‘Oh look at this book that fell from the sky. It is scripture and God never bothered to communicate to man before this.' Is that not funny and unlkely?
-2- They have been miraculously preserved. Anyone objective person who has looked into the scriptures is amazed at what high quality there is in the manuscripts, while non biblical ancient manuscripts rarely have the same quality of preservation, indicating God’s hand in it. Not only so, but many times human powers in the world have tried to exterminate the scriptures but have failed.
-3- The scriptures from genesis to revelation have a begging middle and end. Other books that are proposed to be slotted in have no place to properly fit and contents inside of them contradict contents inside the current scriptures. Not only so but the scripture have a perfect gradual unfolding of one uniform truth that goes beyond human comprehension.
-4- The scripture is unique in that it covers so many subjects without having any internal contradiction.
-5- All scripture speaks in a tone that comes with a full authority of God. Scripture never say this might be the way it is.
-6- Most of the scripture were written during periods of divine attestation from God himself through extra ordinary miraculous displays of his power and acceptance of those persons who wrote, or whose words and actions were written about, in scripture.
-7- Many of those who write the scriptures suffered as martyrs showing a content rejection by the world of the content of the scripture making their unexplainable power and preservation all the more miraculous.
-8- The fact that the scriptures seems to have often overcome the world and turned it upside down even in the face of its attempt to reject them further indicates the internal power of their voice.
So in answer to you question it is simply that the scripture are miraculous and perfect and so those who have believed in them and collected them, carefully preserved them, and they have by this means fallen into our hands and we accept them on the same grounds. Obviously it takes faith in God to believe in scripture. This is the foundation that they assume we will stand on, when falling under there divine and total authority over our lives.
Note: most of this is simply a paraphrase of some arguments found in ‘The reason of faith' by John Owen, Owens Works Volume 4