In Koine Greek, the language the gospels were written in, the word was anamnesis, which coincides with the Jewish word zikkaron, which had significance for the Passover. The idea wasn’t just that you were remembering the Passover, but that you were mystically making present and participating in the original Passover. It wasn’t a re-enactment, but a re-presenting and participation in that first Passover. The fact that Jesus himself uses it for the Eucharist, our new Passover, is important. The implication is not just remembering the past or a particular moment or even just a symbol, but about mystically making present and participation in the original actions on the cross, re-presented at each Eucharist since and first presented (prior to the actual action) at the Last Supper.
And yes, it is his body, as presented at the Last Supper, as it was presented in the Eucharistic Discourse in John 6. It is, literally, the new manna from Heaven, which Jewish tradition expected the Messiah to bring back (see the opening to the Eucharistic Discourse, and I think the manna connection is exceptionally powerful), but it is superior to the old manner. We are not being asked to eat snippets of human flesh and blood, we are being called to take into ourselves the divine in a holy covenant. We are being nurtured by God himself in the manna, in a total self-giving of Himself. The Eucharist also goes back to the idea of the shewbread (bread of the presence) and of course, perhaps most importantly, the Paschal Lamb. The Passover rights required not just the sacrifice of the lamb, but that the sacrificial lamb itself was eaten by the family. That was part of the prescribed ritual of the Passover.
From the writings of Ignatius and the Martyrdom of Polycarp likening martyrdom to the Eucharist, from Saint Justin Martyr describing the Eucharist as Christ’s real flesh and blood, the writings of Irenaeus, etc… This change was understood by the earliest Christians, even if the language wasn’t fully developed. The decisions of the Church at later councils were not new innovations, but statements meant to protect the oldest traditions handed down by the Church, the same way we see the doctrine of the Trinity and the language used to speak about it develop throughout the fourth century as we see the divinity of Christ attacked first by the Arians and then the divinity of the Holy Spirit attacked once the Arian dispute was resolved. The creeds we have today, this refined language on the Trinity, while being the oldest doctrine, was not fully developed or articulated in the first or second century, but was done in response to heresies and challenges the Church faced. Same with the Eucharist.