Given the nature of Hebrew poetry, the phrase "one who shared by bread" should be considered in this case relatively synonymous with "my best friend", and "one I trust completely."
“My friend” is “the man of my šālôm,” someone who should have been committed to my shalom or someone with whom I had a covenanted relationship. The parallel colon offers a complementary description: it is someone I trusted. Verse 9b offers a third description: it is someone who has been in the habit of accepting my hospitality, like a lover (Prov. 9:5) or someone for whom I accept responsibility (Neh. 5:14–18) or a member of my family (Job 42:11; contrast Isa. 4:1) or—more ironically—like an enemy whom I have treated as a friend (Prov. 25:21).
Goldingay, J. (2006). Baker Commentary on the Old Testament: Psalms 1–41. (T. Longman III, Ed.) (Vol. 1, p. 586). Grand Rapids, MI: Baker Academic.
It is less clear whether it can be extrapolated from this single use that anyone who "shared bread" with a man would be considered a close friend of his. For instance, Abraham shared bread with Melchizedek, which many consider to be simply good hospitality. And yet, it is clear from Abraham's offer that it is a sign of peace and friendship. Similarly, peace/fellowship offerings were usually commemorated with a sharing of bread and other provisions (cf. Lev. 7:11f; Joshua 9:12-15). We see also that Boaz's kindness to Ruth includes an offer to eat bread together (Ruth 2:14). David as a sign of peace invites Saul's grandson Mephibosheth to eat bread at his table. And kings ends on a note of hope with Jehoiachin being elevated from prison to the table of the Babylonian king, showing his favor with the king.
Taking then this verse in its historical and canonical context, it does seem to indicate that the sharing of bread together would have been seen in the day as a sign of particular peace and friendship, making the betrayal of one who shared bread with a man particularly grievous as in the case of the psalmist.
The brief answer to your sub-question is that, yes, some people believe this to be fulfilled in some sense in Judas' betrayal of Jesus. This is plain from John 13:18, where Jesus indicates that what is about to happen with Judas is to fulfill Psalm 41:9. So at least Jesus himself (or the author of John, anyway) believed it was fulfilled in him (however that is understood is beyond the scope of the question). Presumably, therefore, many of the early Christians would have seen it this way as well.