When Paul wrote about the many members making up one body in his first letter to the Corinthians, he started that instruction immediately after his admonitions about the Lord’s Supper. Communion implies sharing something in common, which implies that there is more than one to do the sharing.
We learn along with the Corinthians and with the Romans that not any individual has all the spiritual gifts/functions in himself. One part does not make a body (1 Corinthians 12:19), we are a body all together, and God has given us a measure, a portion, so that we can do our part.
This is not like Christ.
Commentators have properly called attention to the difference in respect of measure between Christ and the members of His body. He is “full of grace and truth” (John 1:14), it pleased the Father that “all the fulness should dwell in Him” (Col. 1:19), “in Him are hid all the treasures of wisdom and knowledge”. There is no measure to His endowments. (John Murray, on Romans 12:3-5)
We cannot think more highly of Christ than we ought to think. He is the Head, and from Him all our nourishment comes (Ephesians 5:29-30). He is the fullness of God who fills all in all (Ephesians 1:23). And He is the one we share.
The cup of blessing is a participation in the blood of Christ, we who are many are one body who partake of the one bread, the bread of life Himself (1 Corinthians 10:16-17).