What is the Biblical basis for sharing the cup at communion? In our church everyone usually drinks out of the same cup at communion. Is there a scriptural reason for this?
There are two main reasons:

Most Christians want to think of themselves as continuing the same practices as the early church did and even the very Last Supper itself. Because they shared one cup in the Jewish Passover, so many Christians think it is most appropriate to share one cup now.

There is a passage in 1 Corinthians which describes the emblems of communion as symbols of Christian unity. Though it explicitly only says that the bread is one, it's easy to see how the meaning would also apply to the cup.

1 Corinthians 10:16-17 (NIV): Is not the cup of thanksgiving for which we give thanks a participation in the blood of Christ? And is not the bread that we break a participation in the body of Christ? Because there is one loaf, we, who are many, are one body, for we all share the one loaf.