Today this term is used in various applications. Some use this in discipleship to have one come under them as a covering (which is not biblical).

Biblically Paul did not say it but inferred him being the Corinthians spiritual father, which meant he birthed them by the gospel, not having anything to do with his discipleship.

This is all that it refers to:
1 Cor. 4:15-16 “For though you might have ten thousand instructors in Christ, yet you do not have many fathers; for in Christ Jesus I have begotten you through the gospel. Therefore I urge you, imitate me.”

He later said in 1 Cor. 11:1 “Imitate me, just as I also imitate Christ.”

In other words do not follow him or do what he does unless he is doing it in Christ and he is being a human example for them.