I think it’s fair to say that in Paul’s greeting to Philemon we see things we’re used to seeing in Paul’s greetings, and we see things we’re not so used to seeing in Paul greetings.
We’re used to seeing Paul’s name at the beginning of his letters, this is simply how one wrote letters in the first century. We’re used to seeing Paul include Timothy with himself in greetings, (he does it in 5 of his letters), and he did it here because Philemon had probably met Timothy in Ephesus where Paul was being held in his Roman prison. We’re used to seeing names to whom the letter is addressed in Paul’s greetings, and we’re used to seeing Paul say v3. “Grace to you and peace from God our Father and the Lord Jesus Christ” appears in the greetings of all 13 of his letters, and shows Paul’s affirmation of the deity of Christ by linking the Father with the Son.
We’re not used to Paul addressing letters to individual people.
Of all 13 letters only 9 are addressed to churches while the remaining 4 are addressed to 3 different people: Timothy (received 2), Titus, and Philemon. v2 shows us Philemon was a personal friend of Paul’s in that Paul calls him a beloved fellow worker, which also means Philemon was a minister of the gospel as well. We see Paul mention “Apphia our sister” and “Archippus our fellow soldier” who very well could be Philemon’s wife and son who also knew Paul well. We actually hear of Archippus in Colossians 4:17 where Paul calls him a fellow soldier and encourages him to fulfill the ministry he’s received from the Lord, which shows us that Archippus was undoubtedly one of the leaders in the Colossian church along with Philemon.
We’re not used to Paul identifying himself as a prisoner in his greetings. Out of all his letters the only ones where he doesn’t introduce himself as “Paul, an apostle of Christ Jesus” are Philippians, both letters to the Thessalonians, and here in Philemon. Plus, only here in Philemon do we see Paul identify himself as a prisoner in the greeting. That he does this rather than identifying himself as an apostle is telling and purposeful, and it sets the tone of the whole letter. By beginning this way the letter immediately feels personal, gentle, and more like a friendly appeal than a letter of full-blown apostolic authority. Don’t get me wrong, the letter is still from Paul and Philemon knew that, but Paul is intentionally going out of his way to reduce the feel of ‘command’ here. It also would have reminded Philemon of the severe hardship Paul has had to face in preaching the gospel and in comparison with Paul’s suffering for the gospel the thing Paul is about to ask Philemon to do will seem very small in comparison to what Paul has gone through.
So you see, even in this greeting Paul is already at work, using gospel-logic to prepare Philemon for what he’s about to ask him.