Matthew 1 says,
20But while he thought on these things, behold, the angel of the LORD appeared unto him in a dream, saying, Joseph, thou son of David, fear not to take unto thee Mary thy wife: for that which is conceived in her is of the Holy Ghost.
21And she shall bring forth a son, and thou shalt call his name JESUS: for he shall save his people from their sins.
22Now all this was done, that it might be fulfilled which was spoken of the Lord by the prophet, saying,
23Behold, a virgin shall be with child, and shall bring forth a son, and they shall call his name Emmanuel, which being interpreted is, God with us.
24Then Joseph being raised from sleep did as the angel of the Lord had bidden him, and took unto him his wife:
25And knew her not till she had brought forth her firstborn son: and he called his name JESUS.
In Luke 2:5 it says,
Luke 2:5 (King James Version)
To be taxed with Mary his espoused wife, being great with child.
a footnote in the NET Bible says,
Traditionally, “Mary, his betrothed.” Although often rendered in contemporary English as “Mary, who was engaged to him,” this may give the modern reader a wrong impression, since Jewish marriages in this period were typically arranged marriages. The term ἐμνηστευμένῃ (emnhsteumenh) may suggest that the marriage is not yet consummated, not necessarily that they are not currently married. Some mss read “the betrothed to him wife”; others, simply “his wife.” These readings, though probably not original, may give the right sense.
Upon checking a textual commentary published by the American Bible Society, the evidence for the word “betrothed” is uncertain or ambiguous at best.
So, yes it is possible that the word betrothed is a mistake in our modern texts.
Luke 2:5 (Daniel Marsh edition )
To be taxed with Mary his wife, being great with child.
In Luke chapter 1, we are told that Mary stayed with her cousin for three months until she birthed John.