Checked out the notes on the side of the YouCat. Probably more in the Catechism, but since I'm studying the YouCat now, that's what I'll roll with for an answer
The Christian does not say "My Father" but "Our Father". Even in the secrecy of a closed room, because he knows that in every place, on every occasion, he is a member of one in the same Body.
Pope Benedict XVI, June 6, 2007
That's a little broader than just talking about the Lord's Prayer. I think the pope is saying that we're praying each prayer as the Body of Christ.
We still say the Creed, "I believe" though, because believing is something we can't do for each other. Praying in supplication and for intercession is something we do in communion. Especially in communion with the Church Triumphant in Heaven and for the Church Suffering in Purgatory, but also with the Church Militant here on earth in some mystical way we can't really understand.
In the Lord's Prayer, we say all together "Our Father". So says the Emperor, the beggar the slave, the master. They are all brothers because they have one Father.
St. Augustine of Hippo
And St. Augustine, as always, puts things very succinctly. We also pray with our brothers and sisters here on earth, together. We pray for our brothers even when they're not with us because unlike Cain, we should be about our brother's business.