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.
There are passages, plenty of them, in which the cup containing the fruit of the vine is referred to in the singular, but Luke indicates that this is synecdoche.
In Luke 22:17-20, Jesus starts by directing the gathered disciples to divide up the Cup (meaning its contents), with the later text--in conjunction with the narratives in the other Gospels--indicating that they did not consume it until after eating of the broken bread; so individual containers were employed.
What I gather from this is that both practices are authorized, but neither is mandated beyond the simple fact that each congregation must do one of the two.