This short little letter, delivered by former runaway slave Onesimus to his master, Philemon, shows relational problems between the slave and his master, then how the slave began to relate to Paul (spiritually), with the expectation on Paul's part that the slave's return to his master with this letter would restore the ruptured slave/master relationship due to the slave now having become a Christian.
When Onesimus ran away to Rome, he came across Paul, who explained the good news of Christ to him, resulting in the slave becoming a Christian. This meant that these former strangers were now united spiritually by becoming brothers in the faith. Paul was not related physically to Onesimus! Neither was there any physical relationship between him and Philemon. But all three were now united spiritually - adopted into the family of God by the new (spiritual) birth and hence they were all 'brothers'.
This is how The Companion Bible notes explain this relationship:
"Onesimus is the Latinized form of the Greek Onesimos, which means
'useful', or 'profitable'. He was a slave and, fleeing from his
master, found his way to Rome, where he was, under Paul, led to become
the Lord's freeman, and 'called being a servant (slave)', he cared not
for it, but was willing to return to his master's service, whether to
continue as slave or as 'brother beloved' (vs. 16). From Col. 4:9, we
learn that Paul had sent unto the church at Colosse [where Philemon
seems to have been a believer] Tychicus 'with Onesimus, a faithful and
beloved brother, who is one of you'." (Bullinger, p1820)
When Paul wrote of their brother Onesimus being "useful in the flesh", that meant both useful to Paul (who would fain have kept him in Rome for his service - vs. 13) but especially useful to Philemon as a returned servant. Onesimus was no more Paul's physical brother than he was Philemon's.
The usefulness referred to physical work, or service (in the flesh). But spiritually, "in the Lord", he was now their brother. That is why The Companion Bible rendition of vs. 16 is clearer than the one you are working from. It reads:
"Not now as a servant, but above a servant, a brother beloved,
specially to me, but how much more unto thee, both in the flesh and in
the Lord?"
That is why Paul sent Onesimus back to Philemon, to restore that broken relationship of slave and master ('in the flesh') because now he was above a servant - Onesimus was a beloved brother (spiritually), both to Paul and to Philemon.