This question cannot be answered without assessing what this (whatever) was to be built upon this same (whatever). So what was the "whatever" called the πέτρα--petra--church, and what was the whatever called the ἐκκλησία--ekklēsia--"rock"? It might even help immensely to know what was meant by the κλείς--kleis--"key" (not plural).
First of all, every earthly structure must have a foundation, good or bad. Even the earth, itself, has a foundation as shown in Ezekiel 48:12-13:
Hearken unto me, O Jacob and Israel, my called; I am he; I am the first, I also am the last. Mine hand also hath laid the foundation of the earth, and my right hand hath spanned the heavens: when I call unto them, they stand up together. (My emphasis)
That prophet, Isaiah, spoke of a very special foundation that is connected to some sort of "believing", that the gates of hell shall not prevail against, in Isaiah 28:16-17:
Therefore thus saith the Lord GOD, Behold, I lay in Zion for a foundation a stone, a tried stone, a precious corner stone, a sure foundation: he that believeth shall not make haste. Judgment also will I lay to the line, and righteousness to the plummet: and the hail shall sweep away the refuge of lies, and the waters shall overflow the hiding place. And your covenant with death shall be disannulled, and your agreement with hell shall not stand; when the overflowing scourge shall pass through, then ye shall be trodden down by it. (My emphasis)
All these things did not happen when Israel rebuilt the city and the temple during their captivity. Moreover, Matthew 16's promise was still future. It was not fulfilled that day because Jesus said "I will build my ἐκκλησία".
What then was to be built? Of course, governmental gatherings, whether religious denominatial, or civil, of any sort, in order to lay claim to being this thing that was to be built, must match all the things described in Isaiah 28:16-17, supra. The only chance for such adherence to fulfillment is the belief of individual members of any, or all of the above gatherings. Accordingly, it is possible for πέτρα to be any of Thayer's Greek Lexicon under STRONGS NT 1577: ἐκκλησία.
That belief must be built upon that sure foundation, which is Christ. We know this because Jesus said it was His ἐκκλησία--my church. Although it is Jesus' church, He said it would be built upon "this petra" (rock). First of all, does anyone other than Jesus fit the perfectly cut cornerstone of Isaiah--that stone which the builders rejected--? Strong Defined pétra as follows:
πέτρα pétra, pet'-ra; feminine of the same as G4074; a (mass of) rock (literally or figuratively):—rock.
There are many stones of a temple, but Jesus is that "cut" stone of Daniel 2:45, not the mass rock--not the mountain:
Forasmuch as thou sawest that the stone was cut out of the mountain without hands, and that it brake in pieces the iron, the brass, the clay, the silver, and the gold; the great God hath made known to the king what shall come to pass hereafter: and the dream is certain, and the interpretation thereof sure.
The petra for the ἐκκλησία was clearly tied to a certain belief that had been expressed by Peter. Peter never went around condemning or approving men on earth to be bound in heaven. He, along with the other apostles, were all given the kleis--key (singular) to the kingdom of heaven. No man living at that time, if they did not believe in the coming of Messiah, would ever enter the kingdom of heaven. They would die in their sins and not be resurrected until the second resurrection after the thousand year reign of Messiah was fulfilled, as we see in John 8:21-24:
Then said Jesus again unto them, I go my way, and ye shall seek me, and shall die in your sins: whither I go, ye cannot come. And he said unto them, Ye are from beneath; I am from above: ye are of this world; I am not of this world. I said therefore unto you, that ye shall die in your sins: for if ye believe not that I am he, ye shall die in your sins. (My emphasis)
Many saints believed in Messiah, from Adam, forward. Many Jews believed in their Messiah, from Abraham forward. Most of them have died, but not in their sin, because they believed in their Messiah, even as the believing Jews at the time of of Jesus' ministry as in John 8:31-32:
Then said Jesus to those Jews which believed on him, If ye continue in my word, then are ye my disciples indeed; And ye shall know the truth, and the truth shall make you free. (My emphasis)
Remember, at the time that Jesus called Peter the rock, He also said to them:
Then charged he his disciples that they should tell no man that he was Jesus the Christ. (My emphasis)
The church age had not began. This was still the age of the first 69 weeks of years of Daniel. These weeks of years were those that were determined upon Daniel's people, the Jews--not the church. John, the baptist went only to the Jews. Jesus went only to the Jews. Jesus had not yet finished the work of Salvation--not until He said: *"It is finished."
Therefore, the believers in John 18:32, supra, "shall know" in the future. Jesus gave the gospel of salvation to the twelve, including Paul. They preached that gospel through all the world, beginning at Jerusalem--beginning at Zion (1 Kings 8:1). After calling Simon the Rock because of his belief (therefore also in the future from John 8:31-32), Jesus, Himself, after his death, preached to those believing spirits of all those "saints" who had physically died, from Adam forward, as that Rock, Peter, inform in 1 Peter 3:18-19:
For Christ also hath once suffered for sins, the just for the unjust, that he might bring us to God, being put to death in the flesh, but quickened by the Spirit: By which also he went and preached unto the spirits in prison; Which sometime were disobedient, when once the longsuffering of God waited in the days of Noah, while the ark was a preparing, wherein few, that is, eight souls were saved by water.
That key was the belief in Messiah without seeing Him. Peter and the rest of the apostles were only sent out to preach the gospel after Jesus had ascended up "out of their sight", as detailed in Acts 1:9:
And when he had spoken these things, while they beheld, he was taken up; and a cloud received him out of their sight. (My emphasis)
THEREFORE, the stone is Christ even as the ark was Christ. The Rock is the foundation, that common "belief in Messiah", as 1 Corinthians 3:11 demands:
For other foundation can no man lay than that is laid, which is Jesus Christ. (My emphasis)
So Jesus had to ascend out of their sight so that faith could arise out of the preaching of the Word of God by Peter and the other apostles, as it is written in Romans 10:1:
So then faith cometh by hearing, and hearing by the word of God.
This is crucial because Hebrews 11:6 speaks to all mankind, from Adam to the time of the resurrection of Christ:
But without faith it is impossible to please him: for he that cometh to God must believe that he is, and that he is a rewarder of them that diligently seek him. (My emphasis)
:Peter was given that key to the Kingdom of heaven because Peter was the apostle to the Jews to whom will be given this "kingdom of heaven" on earth through the person of the Son-of-man King of kings, Jesus the Messiah.