In Jesus day there were a number of Jewish sects. The most well known are the Pharisees, Saducees, Essenes, and the Zealots.
These sects had houses of teaching within them .
Within the Pharisees there were two famous rival houses.
They were the House of Shammai(Beit Shammai) and the House of Hillel (founder of the Beit Hillel school). The individuals that these houses were named after were considered the **“father” **of the house.
Jesus warns the apostles not to be fathers to “separate” houses for there is but one house of the Lord. Jesus also says to call no man “teacher” or “master.”
Paul was taught by Gamaliel[Acts22:3], a pharisee from the house of Hillel. Gamaliel was, in fact, the grandson of Hillel. Gamaliel was known as an elder and had the title of “Rabban” which means “Our Master.” This title explains why Jesus also said call no one “master”, for there is but one house and one master of all.
Again, Jesus wanted unity see John 17], and does not want the apostles to set up separate houses or schools of thought. The corollary of this is clearly expressed in Paul’s complaint in 1 Cor 1:10-15 where some are arguing their superiority by being baptized by, and belonging to, Paul vs. Apollos, or Cephas, etc. Paul goes on elsewhere to further warn against factions and divisions, and stresses the importance of one mind and unity of doctrine.
The meaning of “call no man father, teacher, or master” concerns unity and consistency of doctrine and has nothing to do with the commonly used references that are seen elsewhere in scripture or everyday use.
I hope this helps.