"The Son of Man", Jesus' favorite designation of himself, in view of Numbers 23:19-20?

The most common answer to your question is that the what you have found is just another Biblical indication of The Mystery of Faith (i.e. the Trinity). The old testament verse from Numbers indicates that God the Father Almighty is distinct from God the Son. Indeed, "God becoming truly man while remaining fully God" is a common component of short statements of the nature of the Mystery, at least among the Catholic Faith. Differing understandings of how Christ is both human and divine are the essential heart of the Chalcedonian controversy which is one of the largest components of the Great Schism between the East and West. Both East and West, however, agree that Christ was in some sense truly Human and truly Divine; though they attack and insult each other as Arians and Nestorians respectively in neither case are such allegations remotely honest, though they give indication of the direction each feels the other errs.

Christians who profess the heresy of nontrinitarianism would probably say that Jesus isn't God, which explains this set of verses much easier.

There also used to be those, called Docetists, who believed that christ never really came in the flesh but only appeared to be human. The "Son of Man" quotes would thus be facetious or deceptive or ironic or whatever, depending on what motivation you wish to ascribe to God in saying an un-truth. Docetism has hasn't been resurrected as modern heresy for a while, those Muslims commit a modified form of this heresy in saying that Christ was never really crucified, he just created an illusion that appeared to be.

I'm glad you asked for the biblical basis behind the interpretations, because the trinitarian camp, of which I am a part, would look pretty strange otherwise. The fact is, there are a great number of verses which support the trinitarian doctrine to the exclusion of the the two heretical views. Any cursory examination of the Bible should suffice to demonstrate that The Father and the Son and the Holy Spirit are all God, and that God is one. As a brief survey, consider:

"Hear, O Israel, the Lord thy God: The Lord is one" (This appears many, many, many times. It is the great prayer of Judaism. It needs no citation and will be given none.)

John 10:30 "I and my Father are One"

Matthew 28:19 "Go ye therefore, and teach all nations, baptizing them in the name of the Father, and of the Son, and of the Holy Ghost"

2 Corinthians 13:14 "The grace of the Lord Jesus Christ, and the love of God, and the communion of the Holy Ghost, [be] with you all. Amen..."

John 1:1 "In the beginning was the Word, and the Word was with God, and the Word was God"

John 1:14 "And the Word was made flesh, and dwelt among us..."

Colossians 2:9 "For in him dwelleth all the fulness of the Godhead bodily."

John 10:30-36 "I and my Father are one."

However, the Bible was compiled over about the same period the Trinity and by the same people, so this basis is somewhat questionable. The Docetists in particular made use of a gospel which was ultimately rejected by the Church's later ecumenical councils, the pseudepigraphal Gospel of Peter. This text provides the basis for the Docetist's beliefs, and would have been a part of any Bible compiled by them, was it not repudiated by Bishop Serapion of Antioch who is largely responsible for the end of Docetism as a movement. There are a number of other Docetic texts, but Docetism seems to have thus far largely limited itself to material heresy and docetics have generally abandoned their heterodox beliefs when corrected by the Church (or else held other heresies as well, of a better known and named variety).

The non-trinitarian response is, as far as I know, limited to the Jehovah's Witnesses. The Jehovah's Witnesses exclusively use as a Holy Book a document known as the New World Translation, which changes the text of most of the verses of the Bible that would contradict their dogma if it was left as it is the version of the Bible developed by the modern Protestant movement. Their version of the Bible also supplies ample textual support for their beliefs, though it has drawn criticism from others for inaccurately describing itself as a translation, since it clearly is not based off of any credible historical text. The Watchtower, for their part, generally respond to such objections by claiming that their translation is, in fact, consistent with existing other texts.