The concept is the same. The keys to David’s house gave authority and access to the house of David. The “keys” gave access to the saving knowledge of Jesus (of David’s house) which is the gospel for salvation. The “keys” were the knowledge and access to Jesus who is the salvation of those who believed. And no it has nothing to do with the papacy. The Apostles gave everything we need with what they wrote about the saving grace of Jesus and the sacrifice He offered “once for all”. If anything all pastors, teachers, and leaders who bring the gospel message have the keys to heaven. It’s the “good news” of the gospel of Jesus Christ.
If the “keys” were intended for all then why was it that Jesus only gave them to Peter, and not to the other apostles. Jesus used very similar language in addressing the other apostles in Matthew 18, and gave them the power to “bind and loose”, but they did not get the keys. Why would they not receive the keys too? Why was it only Peter?
I cannot see how one can read Matthew 16:18-20 and conclude that the keys were for all.