It's pretty blatant hyperbole, evidenced by the fact that St. Peter and St. Paul both used the term "father" in their letters freely and repeatedly.
In Romans 4:11-12:
11And he received the sign of circumcision as a seal on the righteousness received through faith while he was uncircumcised. Thus he was to be the father of all the uncircumcised who believe, so that to them [also] righteousness might be credited,
12as well as the father of the circumcised who not only are circumcised, but also follow the path of faith that our father Abraham walked while still uncircumcised.
In Romans 9:10:
And not only that, but also when Rebecca had conceived children by one husband, our father Isaac
In 1 Corinthians 4:15:
Even if you should have countless guides to Christ, yet you do not have many fathers, for I became your father in Christ Jesus through the gospel.
In 1 Thessalonians 2:11:
As you know, we treated each one of you as a father treats his children,
In 1 Timothy 5:1:
Do not rebuke an older man, but appeal to him as a father. Treat younger men as brothers,
In Hebrews 12:9:
Besides this, we have had our earthly fathers to discipline us, and we respected them. Should we not [then] submit all the more to the Father of spirits and live?
In James 2:21:
Was not Abraham our father justified by works when he offered his son Isaac upon the altar?
In 1 John 2:13:
I am writing to you, fathers, because you know him who is from the beginning.
I am writing to you, young men, because you have conquered the evil one.
St. Paul also used it in Acts 22:1:
“My brothers and fathers, listen to what I am about to say to you in my defense.”
As well as St. Stephen in Acts 7:2:
And he replied, “My brothers and fathers, listen. The God of glory appeared to our father Abraham while he was in Mesopotamia, before he had settled in Haran,
The list of references to the apostles calling people (or themselves) fathers continues on from there.
The dead giveaway though is in Matthew 19:18-19, when Jesus himself makes mention of worldly fathers in his recitation of the commandments:
18He asked him, “Which ones?” And Jesus replied, “ ‘You shall not kill; you shall not commit adultery; you shall not steal; you shall not bear false witness;
19honor your father and your mother’; and ‘you shall love your neighbor as yourself.’”
Jesus was known for using hyperbole as a rhetorical device. Consider Matthew 5:29-30:
29If your right eye causes you to sin, tear it out and throw it away.s It is better for you to lose one of your members than to have your whole body thrown into Gehenna.
30And if your right hand causes you to sin, cut it off and throw it away. It is better for you to lose one of your members than to have your whole body go into Gehenna.
Obviously, Jesus doesn't mean for you to pluck your eye out or cut your hand off.
So if it's not meant to be taken literally, what was the point Jesus was trying to convey? For that, I turn to an essay on Catholic Answers, which was declared nihil obstat by the Censor Librorum of the Archdiocese of San Diego:
Jesus is not forbidding us to call men "fathers" who actually are such—either literally or spiritually. [...] To refer to such people as fathers is only to acknowledge the truth, and Jesus is not against that. He is warning people against inaccurately attributing fatherhood—or a particular kind or degree of fatherhood—to those who do not have it.
This was also a temptation in the Jewish world of Jesus’ day, when famous rabbinical leaders, especially those who founded important schools, such as Hillel and Shammai, were highly exalted by their disciples. It is this elevation of an individual man—the formation of a "cult of personality" around him—of which Jesus is speaking when he warns against attributing to someone an undue role as master, father, or teacher.