The Greek reads of the union with the body and blood of Christ, not of the fellowship of believers.
The blood of Jesus The Christ was shed on that Cross to atone for you and me, to cover our sin nature with His sinless Nature, so when satan accuses in the Throne Room, God looks toward the Laws and sees only the sinless blood fo Christ.
Do you know why God gace the Laws of Moses? Do you realize that they reflect the nature of GOD, the character of The Righteous GOD. THAT nature is impossible for us to be without first He cleanses us then imparts His Holy Spirit into our huamns pirit. Do we live thereafter sinless lives? Not even. But we have an 'advocate', Jesus Christ The Righteous, interceding for us continually before God. He gave His sinless blood for us ONCE, for all time, reaching back to Adam and forward to you and me.
The blood of Christ is imparted to those who had faith long before the Last Supper. But how can that be with the assertion of Catholicism that the Eucharist is the way to receive Christ's blood via the gaping mouths at a catholic altar?
Reading it in Greek, AND IN CONTEXT, no it does not.
Paul uses the example of Israel to make the same argument in the next verse:
18 Look at the nation Israel; are not those who eat the sacrifices sharers in the altar?
Paul goes through extensive teaching in this section that believers should not participate in idolatry. The Lord’s Supper and Israel are his examples.
“Paul’s line of reasoning was proceeding as follows. Christians who eat the bread at the Lord’s Supper thereby express their solidarity with one another and with Christ. Likewise Jews who ate the meat of animals offered in the sacrifices of Judaism expressed their solidarity with one another and with God. Therefore Christians who eat the meat offered to pagan gods as part of pagan worship express their solidarity with pagans and with the pagan deities.”
(Tom Constable)