This is a very loaded passage because there is a big difference between having a religious identity and 'learning Christ' in a new nature.
"But that is not the way you learned Christ! - assuming that you have heard about him and were taught in him, as the truth is in Jesus, to put off your old self, which belongs to your former manner of life and is corrupt through deceitful desires, and to be renewed in the spirit of your minds, and to put on the new self, created after the likeness of God in true righteousness and holiness. Therefore, having put away falsehood, let each one of you speak the truth with his neighbor, for we are members one of another." - Ephesians 4:20-25 ESV
There is truth in Jesus that can't be had in a old identity. This truth is new identity truth. Jesus renamed Simon and called him Peter. He renames us also and in that new name we learn of him. We may answer to the name our parents gave us but we don't have to live in the nature that they gave us. We have a new nature.
"Do not lie to one another, seeing that you have put off the old self with its practices and have put on the new self, which is being renewed in knowledge after the image of its creator." - Colossians 3:9-10 ESV
This new self can speak the truth in love if it is taught to speak in love. The parent teaches the child to say 'please' and 'thank you'. When we learn Christ we learn honest in a new identity. We learn unity in holiness. We learn Agape (self sacrificing) love. It isn't automatic. The capacity is there but it must be taught.
Jesus and the disciples called Simon by his new name and gave him a new identity. People who didn't walk with Christ but knew this man only knew him by his old name. In time Simon became Peter in his experience even though it only took a moment for Jesus to give him his new name.
Anyone can get a new name in the world. Cults do it. Entertainers reinvent themselves every few years to showcase an old talent in a new light. Jobs call it a title. Universities give out degrees that add to your name. But when God does it in Jesus it is the description of a new nature that only He has the power to give. It isn't just a fresh start. It is a declaration of citizenship.
Jesus renamed Simon but he didn't do that with all the disciples. The new name was a tool to reveal a new nature to Peter. All the disciples had a new nature given to them. God is going to use these tools to show us what He has done but they are handcrafted to our experience.
We may answer to the name we have had since our youth but lets live learning Christ in the name He has named us. He calls us free and forgiven. He calls us family. Speaking the truth in love becomes the way we communicate as we live in this new life.