Jesus and his disciples were celebrating Passover when they had the Last Supper, recorded in Matthew, Mark and Luke. There Jesus said, “this is the blood of the new covenant” (Mt 26:28). The KJV translators chose the word “testament” over “covenant”, though in this context, it should probably be translated as “covenant”. This was celebrated before His death, when the disciples were un-regenerated, still under law, hence keeping the Passover on a Sabbath.
In contrast, the Church breaks bread on the first day of week (Acts 20:17) (signifying the 8th day, associated with resurrection, the New Heavens and New Earth, and the New Creation).
Additionally, there is new revelation associated with the Lord’s Table that was given to Paul. We discover that we are the Body, having become one bread and one loaf (1 Cor 10:17). As His Body, we are standing in our position in His resurrection in Christ’s stead as His representatives, “showing forth His death until He comes” as a testimony (2 Cor 11:26).
The Lord’s table that the Church celebrated was a fellowship as the Body in the blood and flesh of Christ (1 Cor 10:16) as heirs, risen with Christ, testifying of the benefits that were received as an inheritance because of His death and resurrection.
Keep in mind that Jesus’ death and His blood were the basis of both the Everlasting (Hebrews 13:20) and the New Covenants (Matthew 26:28). While the New Covenant is said to be made with the House of Israel and with Judah (Jer 31:31; Heb 8:8), the Everlasting Covenant is made between the Father and the Son to make Him the Shepherd of the Sheep. The same blood, the blood of Christ, secures both. The blood does not belong exclusively to one or the other, He didn’t die twice, once for Israel and once for gentile believers, He died once for all (Hebrews 10:12). It is the blood of Christ that gives value and strength to the covenant.