‘Name’ had more than one meaning in biblical times; a point lost on most people today. We are inclined to think of what’s written on our birth certificates as our name/s, so your name may be Charles Mbogo, and you are asking if God the Father has a similar personal name designation. While it is true that ‘God’ is a general title and not a personal name, there are ways of differentiating God the Father from all the other gods one hears of in this world. “Father” is one such designation that differentiates him, as is “Almighty”.
It also needs to be borne in mind that coming in the ‘name’ of someone means coming in that one’s authority, as representative, as speaking what that one would speak if personally present, and doing what that one would do if personally present. Ambassadors have that authority, when representing their nation’s government, or king or queen, while living in a different country. They come, speak, and act “in the name of” their appointed government, king or queen. That is their authority.
Critical to grasping what Jesus meant by coming in the name of the Father, and of how the Jews he spoke to understood this, is to see the context of passages such as John chapter 5 of which you only quote two verses. Consider first what led to the debate about Jesus and in whose name he came, and it soon becomes apparent what he meant by coming in the name of the Father, and why God’s name, ‘Father,’ is so significant.
The Jews had taken offense at Jesus healing a lame man on the Sabbath, to the extent that they wanted to kill Jesus (5:16 The Companion Bible). Jesus told them, “My Father worketh hitherto, and I work.” “Therefore the Jews sought the more to kill him, because he not only had broken the sabbath, but said also that God was his Father, making himself equal with God.” (5:17-18) That is a crucial point so often overlooked by readers of the text. It shows the unique significance of calling God “Father”. For a Jewish man like Jesus to do that meant he was making himself equal with God.
This outraged his critics. So, Jesus explained that he, the Son of God, could do nothing of himself, but only what he saw the Father do – that the Son does what the Father does. He even went so far as to say that the Father had given him authority to raise, and judge, the dead “so that all men should honour the Son, even as they honour the Father. He that honoureth not the Son honoureth not the Father which has sent him.” (5:19-23)
Now, his critics were used to rabbis coming in the name of others of great authority, and they accepted them on the basis of knowing the ones whose names they claimed as their authority. There was a tradition of rabbis listing various acclaimed (and usually long-dead) rabbis whose work they believed and promoted, continuing the ‘line’ of teaching of the original rabbi whose school they represented. This was what Jesus’ critics were expecting him to list – human religious teachers they already knew of. If he came in their ‘name’ [authority] they would accept him. But Jesus ignored all the middle-men. He went right to the top – to God the Father, personally! He came in the name of the Father! That must have sent them into fits of apoplexy.
Now we can grasp the significance of verses 43 and 44, which you quoted. It also lends new significance to his words in verses 31-44, that Christ does not receive honour from men, though they did. As Matthew Henry’s comment states, he came not to seek his own glory.
“…they obstinately opposed him who was the true Messias (v. 43)… Those
are false prophets who come in their own name, who run without being
sent… They therefore slighted and undervalued Christ because they
admired and overvalued themselves… He did not covet nor court the
applause of men, did not in the least affect that worldly pomp and
splendour in which the carnal Jews expected their Messiah to appear.”
In Judaism, certain rabbis came to be so acclaimed that anyone promoting his school of teaching would be accepted. The schools of Shammai and Hillel were current in Jesus’ day. Had he cited them as his authority, he could have been accepted. (Yet, even though his account in Luke 16:13-31 agreed with those schools on that point, he was hated for saying it because it was an expose of the love of money the Pharisees had.) Then, long after Christ, for example, rabbi Maimonides (1135-1204) is still, to this day, upheld as an authority for establishing the Thirteen Principles of the Faith. Again, a later and different religious tradition relies heavily on tracing a line of teachers to authenticate their ‘Hadith’ as almost equivalent in authority to their Qur’an. These ‘traditions’ have to have a traceable line going back to Mohammed, then they will be accepted, and they are considered to be very important for all Muslims to believe. [Sources: The Encyclopedia of World Faiths edited by Peter Bishop & Michael Darton, pp37-38 & Islam, the way of submission, by Solomon Nigosian, Crucible, pp 122-129]
Finally, Jesus culminated his exquisite case by calling himself the I AM (John 8:54-59). Whereas you said, “We know that God manifested Himself as I AM WHO I AM in the OT. Jesus doesn't talk much about that name,” the astounding fact is that he only needed to call himself by that divine name once, and he nearly got stoned to death for it. That is the power and the authority of taking the very name of God upon himself. And his critics knew it, for they knew there was only one God and Father, and that Jesus came in the unique name of God the Father, being his Son.