Be Ye Not Called Masters
Here is Jesus' teaching in context:
Matthew 23:1-12 (DRB) Then Jesus spoke to the multitudes and to his disciples, 2 Saying: The scribes and the Pharisees have sitten on the chair of Moses. 3 All things therefore whatsoever they shall say to you, observe and do: but according to their works do ye not; for they say, and do not. 4 For they bind heavy and insupportable burdens, and lay them on men's shoulders; but with a finger of their own they will not move them. 5 And all their works they do for to be seen of men. For they make their phylacteries broad, and enlarge their fringes. 6 And they love the first places at feasts, and the first chairs in the synagogues, 7 And salutations in the market place, and to be called by men, Rabbi. 8 But be not you called Rabbi. For one is your master; and all you are brethren. 9 And call none your father upon earth; for one is your father, who is in heaven. 10 Neither be ye called masters; for one is your master, Christ. 11 He that is the greatest among you shall be your servant. 12 And whosoever shall exalt himself shall be humbled: and he that shall humble himself shall be exalted.
First, it's important to read a text in the context (accompanying text) in which it was being said. Someone can interpret, "I and the Father are one," to mean that Jesus is the Father, just as they do with, "Whoever has seen me has seen the Father." On the other hand, one can, and people do, also interpret it to mean that Jesus shares one nature and mission with the Father, and that the Son is "the very expression of [the Father's] person." All this to say, words must not be taken on their own, in isolation (leaving us with a kind of 'free-for-all' of interpretive freedom, rendering the text useless practically speaking). We must take into account the religious, geographical, political - all - factors relevant to how the original hearers would have understood what was being said. More important than that, we must at least respect the immediate literary context (the surrounding teaching, what set of teachings and on what the teaching belongs to, etc.).
In this particular case, Jesus is cautioning His disciples against seeking, taking delight in, and vainly hoarding up, the respect of subjects and people of lesser status or importance than themselves. So right off the bat, St. - "I am nothing;" "I am not worthy even to be called an apostle" - Paul, is not going for that whatsoever: to state that he falls behind even the chiefest apostles in nothing is not vainglory, but a reminder that he has the authority of an apostle, and was commissioned by the very same Lord as they (Gal. 1:1, 12). His humility both here and elsewhere - the context of Paul as a writer - assures us of this fact (cf. 1 Cor. 1:12-14).
Call No Man Your Father Upon Earth
This is, as is immediately clear, hyperbolic. Jesus identifies people's biological father as their "father" countless times (Mt. 10:35, 9:5, etc. cf. Heb. 12:9), and hence it can't be morally broken or wrong - i.e spoken against in the above teaching - to identify such persons as what Jesus says they are Himself: that would be direct evidence of misinterpretation of His words (if He Himself precludes that interpretation).
Again, He also uses "father" in a spiritual sense Himself, when he says, "father Abraham" (Lk. 16:24). Clearly Abraham was, for the Jews, leader, exemplar of the Jewish faith, authority, in addition to just bearing Isaac and Jacob: he is, in fact, father of all the just departed, Jesus calling the spiritual abode of the dead before he came, "the bosom of Abraham" specifically.
For example, Jesus also taught:
Matthew 3:9 (DRB) And think not to say within yourselves, We have Abraham for our father. For I tell you that God is able of these stones to raise up children to Abraham.
This is analogous to the passage in question: 'don't call [this man] your father.' However, in this context, He clearly means, 'don't boast about your Abrahamic heritige, as if this guarantees you anything: true sons of Abraham for whom being the son of Abraham counts, bear evidence of his faith' - not that it is wrong to be proud of being born Jewish, a son of father Abraham, which may legitimately be taken pleasure in as a good gift (Rom. 5:9). If God is able to give Abraham sons which are sons by virtue of sharing his faith, and not by virtue of biology, then what are they to call Abraham except, "father Abraham?"
This isn't about the words used, or even titles. Is "the apostle Paul"or "St. Paul" a bad phrase, merely because of the fact that it acknowledges the goodness of the apostolic office, or that he is holy/in heaven? Is it scandalous? Clearly not. Neither can calling your biological or spiritual father "father" in that sense be wrong. Jesus is hyperbolically teaching that we should have nothing to do with the entire trap of pride, of which being called by special titles is a chief example.
Jesus is forbidding seeing men as you true father or origin-point as wrong, inasmuch as to "call" something some thing is a way of saying, "consider" as well as literally "call" (e.g. Mt. 1:23). After all, are students never to call their teachers ("kathegetai"), "Teacher?" as He teaches here? He obviously means for us to understand we are to see no one as an end in themselves: don't follow Rabbis as if they are God.
St. Paul's Being 'Father'
There is clearly no meaningful distinction between fathering someone (male begetting), and being called father (especially since it's in the context of "many tutors") inasmuch as to be father in any sense, is to be legitimately called such in that sense.
The word St. Paul uses is "I have begotten" (egennisa) but the context fixes the meaning of that begetting to, "have become father to," as in spiritual adoption or begetting:
1 Corinthians 4:14-16 (DRB) I write not these things to confound you; but I admonish you as my dearest children. 15 For if you have ten thousand instructors in Christ, yet not many fathers. For in Christ Jesus, by the gospel, I have begotten you. 16 Wherefore I beseech you, be ye followers of me, as I also am of Christ.
There is a healthy, balanced view of spiritual fatherhood, or calling people, "master," or "teacher" (such as students, or servants), and the prideful, distorted view taught against by Jesus, and against which he warns His disciples.