No other Church can legitimately make this claim.
Another thought:
There is no other church.
There is only THE Church and it is under the headship of Christ Jesus who was crucified for us all.
"Be completely humble and gentle; be patient, bearing with one another in love. Make every effort to keep the unity of the Spirit through the bond of peace.There is one body and one Spirit, just as you were called to one hope when you were called; one Lord, one faith, one baptism; one God and Father of all, who is over all and through all and in all."
And this Church, the one true Ecclesia, the people of God; how will they be known?
I cannot find anywhere in scripture that says anything along the lines of: “they shall be known by their organisation and by their apostolic succession.”
Scripture does however make clear in plain language how the body of Christ will be known:
“by their fruit they shall be known.” and, "the fruit of the Spirit is love, joy, peace, forbearance, kindness, goodness, faithfulness, gentleness and self-control."
"He said to them, “Go into all the world and preach the gospel to all creation. Whoever believes and is baptized will be saved, but whoever does not believe will be condemned. And these signs will accompany those who believe: In my name they will drive out demons; they will speak in new tongues; they will pick up snakes with their hands; and when they drink deadly poison, it will not hurt them at all; they will place their hands on sick people, and they will get well.”"
"Therefore, as God’s chosen people, holy and dearly loved, clothe yourselves with compassion, kindness, humility, gentleness and patience. Bear with each other and forgive one another if any of you has a grievance against someone. Forgive as the Lord forgave you. And over all these virtues put on love, which binds them all together in perfect unity."
As for issues of Papal primacy and the apostle Peter:
"An argument started among the disciples as to which of them would be the greatest (an argument that continues to this day).
Jesus, knowing their thoughts, took a little child and had him stand beside him. Then he said to them, “Whoever welcomes this little child in my name welcomes me; and whoever welcomes me welcomes the one who sent me. For it is the one who is least among you all who is the greatest.”
“Master,” said John, “we saw someone driving out demons in your name and we tried to stop him, because he is not one of us.”
“Do not stop him,” Jesus said, “for whoever is not against you is for you.”"
Paul wrote: "I appeal to you, brothers and sisters, in the name of our Lord Jesus Christ, that all of you agree with one another in what you say and that there be no divisions among you, but that you be perfectly united in mind and thought. My brothers and sisters, some from Chloe’s household have informed me that there are quarrels among you. What I mean is this: One of you says, “I follow Paul”; another, “I follow Apollos”; another, “I follow Cephas”; still another, “I follow Christ.”
Is Christ divided? Was Paul crucified for you? Were you baptized in the name of Paul? I thank God that I did not baptize any of you except Crispus and Gaius, so no one can say that you were baptized in my name. (Yes, I also baptized the household of Stephanas; beyond that, I don’t remember if I baptized anyone else.) For Christ did not send me to baptize, but to preach the gospel—not with wisdom and eloquence, lest the cross of Christ be emptied of its power.
Brothers and sisters, I could not address you as people who live by the Spirit but as people who are still worldly—mere infants in Christ. I gave you milk, not solid food, for you were not yet ready for it. Indeed, you are still not ready. You are still worldly. For since there is jealousy and quarreling among you, are you not worldly? Are you not acting like mere humans? For when one says, “I follow Paul,” and another, “I follow Apollos,” are you not mere human beings?
What, after all, is Apollos? And what is Paul? Only servants, through whom you came to believe—as the Lord has assigned to each his task. I planted the seed, Apollos watered it, but God has been making it grow. So neither the one who plants nor the one who waters is anything, but only God, who makes things grow. The one who plants and the one who waters have one purpose, and they will each be rewarded according to their own labor. For we are co-workers in God’s service; you are God’s field, God’s building.
So then, no more boasting about human leaders! All things are yours, whether Paul or Apollos or Cephas or the world or life or death or the present or the future—all are yours, and you are of Christ, and Christ is of God."