At the School of Theology, all of our writing has to conform to the standards of the eighth edition of Kate Turabian's A Manual for Writers of Research Papers, Theses and Dissertations. The manual is 448 pages long, and the whole thing is about avoiding plagiarism. Section 17.5.2 is dedicated to citing "The Bible and Other Sacred Works."
So, yes... it is possible for a person to plagiarize the Bible itself, if that person copies something from the Bible and claims that she wrote it herself.
And, yes... there are standards for citation in place to avoid plagiarizing the Bible, or any other sacred literature.
Other organizations may use other manuals. The "hard sciences" use the APA style.
Love,
Steve