What was Esther’s relationship to Mordecai? Do we know what was Esther's relationship to Mordecai, based on the Jewish and Christian tradition?
There are two views on the relationship between Mordecai and Esther.  


That she was his wife, is shown in one translation of the Septuagint.  
That she was his adopted daughter, is shown in the Hebrew (per Josephus) and translated Masoretic texts



  Esther 2:7 Septuagint
  And he had a foster child, daughter of Aminadab his father's brother, and her name [was] Esther; and when her parents were dead, he brought her up for a wife for himself: and the damsel was beautiful.
  
  Esther 2:7 Masoretic
  And he brought up Hadassah, that is, Esther, his uncle's daughter: for she had neither father nor mother, and the maid was fair and beautiful; whom Mordecai, when her father and mother were dead, took for his own daughter.


So, was Esther Mordecai’s wife or daughter?  We will see that the “for himself” in the Septuagint, is a misunderstanding.  Mordecai adopted Esther for his own daughter and yes to bring her to be a wife.

To return to Esther, the king sent out a command that the best virgins were to come to him (2:2-4).  If Mordecai had married Esther, this would exclude her; but, she was included in the search (2:8).  This tells us clearly that they were not married, though Esther was of marriageable age.  Mordecai was bringing her up to be married, though not to him.

We can also find this same line of thought in Josephus.  He was aware of the Hebrew text.  The Greek Septuagint appears to be a mistranslation.


  Preface to Antiquities
  2. Now I have undertaken the present work, as thinking it will appear to all the Greeks worthy of their study; for it will contain all our antiquities, and the constitution of our government, as interpreted out of the Hebrew Scriptures.
  http://www.biblestudytools.com/history/flavius-josephus/antiquities-jews/preface/chapter-1.html
  
  Antiquities Book 11 Chapter 6 Paragraph 2
  So when a great number of these virgins were gathered together, there was found a damsel in Babylon, whose parents were both dead, and she was brought up with her uncle Mordecai, for that was her uncle's name.
  http://www.biblestudytools.com/history/flavius-josephus/antiquities-jews/book-11/chapter-6.html


So, in the Jewish and Christian tradition, the answer is Esther was Mordecai’s adopted daughter.  He was bringing her up in the norms of the day.  She was raised a virgin, was never Mordecai’s wife, and became married to the king.

Notes
For an outline that the Septuagint was variously translated see https://en.wikipedia.org/wiki/Septuagint

PS I suspect the "married" idea of "old guardian" to "young virgin" Esther 2:7 translation of the Septuagint was made later to conform it to the later tradition from the Infancy Gospel of James of its tale of the "old guardian" and "young virgin".