Had Jesus formally been given authority to preach in the synagogue?

Lk 4:14-15 NIV Jesus returned to Galilee in the power of the Spirit, and news about him spread through the whole countryside. He was teaching in their synagogues, and everyone praised him.

The authority? In the power of the Spirit. He shapes the situation, not things happening by chance, including whatever happening in the synagogue. Divine intervention, though not explicitly spelt out, is what I see in this chapter.

From the commentary on St. Luke 4:16 by Conelius à Lapide, S.J.:

And stood up for to read. It was (and still is) the custom among the Jews that each one should read the Hebrew books of Holy Scripture in the synagogue on the Sabbath-day, both that he might learn the law of God from it, and also that he might be stirred up to the worship, love, and service of God. Moreover, it was the part of the Rabbin and the teachers, such as Jesus was, to read the Holy Scripture publicly, to interpret it, to preach, and to teach.

St. Ambrose commentates (recorded in St. Thomas Aquinas's Catena Aurea on Luke 4):

The Lord in every thing so humbled Himself to obedience, that He did not despise even the office of a reader, as it follows, And he rose up to read, and there was delivered to him the book, &c.

Quite amazing! As Origen says:

He opens not the book by chance, and finds a chapter containing a prophecy of Himself, but by the providence of God.

Jesus was addressed as Rabbi – which means “my teacher” (Matthew 26:49). In Judaism, a rabbi is a teacher of Torah. Here is a brief extract from a Jewish article on the evolution of the Rabbi, showing how authority to preach in the Synagogue was granted orally during the first century:

“For most of Jewish history, the primary qualification for this title was sufficient learning to render decisions in Jewish law. In the earliest stages of Jewish history, the ability to rule in matters of Jewish law was handed down orally from teacher to student in an unbroken lineage going back to Moses. At God’s command, Moses ordains Joshua as his successor to lead the Jewish people and render judgments, a process he effects by laying his hands upon him.” https://www.myjewishlearning.com/article/rabbi-teacher-preacher-judge-but-not-priest/

Jesus was qualified to teach in the Synagogue of his home town, Nazareth because he was a religiously observant Jew and he was beyond the age of accountability or coming into maturity. He wore tzitzit (tassles) on His clothing (Luke 8:44; Matthew 14:36) to serve as a reminder of the commandments (Numbers 15:37-39). He observed Passover (John 2:13) and went up to Jerusalem (Deuteronomy 16:16) on this very important Jewish pilgrimage feast day. He observed Succoth, or the feast of tabernacles (John 7:2, 10) and went up to Jerusalem (John 7:14) as required in the Torah. He also observed Hanukkah, the festival of lights (John 10:22) and probably Rosh Hashanah, the feast of trumpets (John 5:1), going up to Jerusalem on both those occasions as well.

Many synagogue customs can be observed in Luke 4:16-21. First, the meeting took place on the Sabbath (Saturday). Second, Jesus stood to read. Third, He read from a scroll. Even today, scrolls are found in synagogues and are used for weekly readings (see also Acts 15:21). When finished with His reading Jesus sat down to teach, another synagogue tradition. Upon arriving in a new community the Apostle Paul (formerly known as Saul of Tarsus) would show up at the synagogue and request to speak. He definitely had the necessary credentials:

“I am a Jew, born in Tarsus in Cilicia, but brought up in this city, educated at the feet of Gamaliel according to the strict manner of the law of our fathers, being zealous for God as all of you are this day” (Acts 22:3).

Like Paul, Jesus also had the necessary credentials to teach in the Synagogue as a Rabbi:

Jesus was a Jew: “The book of the genealogy of Jesus Christ, the son of David, the son of Abraham” (Matthew 1:1).

Jesus was a religiously observant Jew. Both of Jesus' parents had “done everything required by the Law of the Lord” (Luke 2:39). His relatives, Zechariah and Elizabeth, were also Torah-observant Jews (Luke 1:6).

“Every year his parents went to Jerusalem for the Feast of the Passover. When he was twelve years old, they went up to the Feast, according to the custom... After three days they found him in the temple courts, sitting among the teachers, listening to them and asking questions. Everyone who heard him was amazed at his understanding and his answers” (Luke 2:41-52)

Jesus taught in the Jewish Temple in Jerusalem: “And every day he was teaching in the temple” (Luke 21:37). If He were not a Jew, His going into that part of the Temple would simply not have been allowed (Acts 21:28-30).

There follows a brief extract from a Catholic source - Jesus Christ Is Recognized as the Divine Rabb. This is an inspirational and uplifting insight into how Jesus taught, and ultimately, proving that his authorisation came from God:

“Jesus Christ is the Divine Rabbi (Teacher). He is Divine, the “Son of God” (Luke 1:32), and the Rabbi (Hebrew, rab, meaning “great”). Jesus acknowledges His role, saying, “You call me Teacher and Lord; and you are right, for so I am” (John 13:13). His whole life is “continual teaching” (CCC 561) for Jesus Christ is the Divine Rabbi. Jesus was recognized as a Divine Rabbi among the Jewish people and leaders (CCC 581). At 12, Jesus amazed the rabbis in the Temple (Luke 2:46-47). His primary title with His disciples was “Rabbi” or “Teacher” (John 1:38; Mark 4:38). He goes “about all the cities and villages, teaching in their synagogues” (Matt 4:23; 9:35; 11:1; Mark 6:2) and in the Temple (Mark 14:49). His teachings “astonish” the crowds (Matt 7:48; 22:33; Mark 1:22; 6:2). He is recognized as a great teacher by the scribes/Pharisees (Matt 8:19; 9:11; 12:38), the chief priests (Luke 20:21) and the Sadducees (Matt 22:23-24). After the Resurrection, Mary cries, “Rabboni” (John 20:16), underscoring the Jesus’ eternal role as Divine Teacher.” http://www.catholicmannight.com/uncategorized/jesus-christ-divine-rabbi/