Before looking at the Quran, let's see what the Bible has to say about the characters in question.
In the Old Testament, Amram is the father of Moses, Miriam, and Aaron (Exodus 6:20, 15:20). He is a minor character, whose life we know next to nothing about.
In the New Testament, Mary, the mother of Jesus, is descended from Judah (Luke 3:26, Hebrews 7:14). She has a cousin named Elizabeth (Luke 1:36). Elizabeth's husband is a priest named Zachariah. Their son is John the Baptist (Luke 1).
The following passage in Surah 19 is in context of Mary giving birth to Jesus, returning from the desert with Him to her family:
Surah 19:27-28
27 Then she returned to her people, carrying him. They said ˹in shock˺, “O Mary! You have certainly done a horrible thing!
28 O sister of Aaron! Your father was not an indecent man, nor was your mother unchaste.”
Notice that she is called the "sister of Aaron". That detail is found nowhere in the Bible. However, there is a "Mary" in the Bible with a brother named Aaron. The issue is, that "Mary" lived about 1,500 years before Mary the mother of Jesus, and in English, we call her "Miriam". Since the two names are the same in Arabic (Maryam), and Muhammad barely knew anything about the Bible, he conflated them.
This is established more clearly by looking at a passage in Surah 3:
Surah 3:33-39
33 Indeed, Allah chose Adam, Noah, the family of Abraham, and the family of ’Imrân above all people ˹of their time˺.
34 They are descendants of one another. And Allah is All-Hearing, All-Knowing.
35 ˹Remember˺ when the wife of ’Imrân said, “My Lord! I dedicate what is in my womb entirely to Your service, so accept it from me. You ˹alone˺ are truly the All-Hearing, All-Knowing.”
36 When she delivered, she said, “My Lord! I have given birth to a girl,”—and Allah fully knew what she had delivered—“and the male is not like the female. I have named her Mary, and I seek Your protection for her and her offspring from Satan, the accursed.”
37 So her Lord accepted her graciously and blessed her with a pleasant upbringing—entrusting her to the care of Zachariah. Whenever Zachariah visited her in the sanctuary, he found her supplied with provisions. He exclaimed, “O Mary! Where did this come from?” She replied, “It is from Allah. Surely Allah provides for whoever He wills without limit.”
38 Then and there Zachariah prayed to his Lord, saying, “My Lord! Grant me—by your grace—righteous offspring. You are certainly the Hearer of ˹all˺ prayers.”
39 So the angels called out to him while he stood praying in the sanctuary, “Allah gives you good news of ˹the birth of˺ John who will confirm the Word of Allah and will be a great leader, chaste, and a prophet among the righteous.”
Here, Mary the mother of Jesus is described as the daughter of Imran - the Arabic version of Amram - and given to Zachariah, the father of John the Baptist. While John the Baptist and Zachariah are certainly Biblical contemporaries of Mary, we are not told anything in the Bible about someone named Imran in relation to Mary the mother of Jesus.
However, there is a "Mary" in the Bible who has a father named "Imran". It's the same Mary who has a brother named Aaron, and once again, she lived 1,500 years before Mary the mother of Jesus, Zachariah, and John the Baptist. Muhammad genuinely did conflate the two Marys.
This mistake is repeated in Surah 66:
Surah 66:12
12 ˹There is˺ also ˹the example of˺ Mary, the daughter of 'Imrân, who guarded her chastity, so We breathed into her ˹womb˺ through Our angel ˹Gabriel˺. She testified to the words of her Lord and His Scriptures, and was one of the ˹sincerely˺ devout.
Mary is also called in the "daughter of Imran" in Sahih al-Bukhari 3411, 3432-34, 3769, 5418, Sahih Muslim 2431, 2527c, and Jami at-Tirmidhi 1834. Nothing in church history, or the Bible, corroborates that her father was named Imran or Amram. In fact, looking at Christ's genealogies indicates that her father was named "Heli".
During his life, this conflation was brought to Muhammad by one of his companions, as we read:
Sahih Muslim 2135
Mughira b. Shu'ba reported:
When I came to Najran, they (the Christians of Najran) asked me: You read" O sister of Harun" (i. e. Hadrat Maryam) in the Qur'an, whereas Moses was born much before Jesus. When I came back to Allah's Messenger (ﷺ) I asked him about that, whereupon he said: The (people of the old age) used to give names (to their persons) after the names of Apostles and pious persons who had gone before them.
First, recognize that this is the canonical Muslim response to this error in the Quran. No one else can give any other explanation, because that would be arguing against what Muhammad says, and suggesting a different, or "better" solution than the one he gave.
So, what does Muhammad say? He says that, in essence, Mary the mother of Jesus, and her father, and her brother - who we have no Biblical record of even existing - all had the same names as the family of Moses, which was a common practice at the time.
Proof? No proof. Scripture? No Scripture. There is no instance anywhere in Scripture of an entire family taking the names of another family from 1,500 years earlier. This is the definition of ad-hoc.
Firstly, as already mentioned, there is no record of Mary having a brother in any Scripture, let alone one named Aaron. If this was referring to her descent from Aaron, we would expect to see "daughter of Aaron", which is how that is denoted in Scripture, and definitely not "sister of Aaron".
Secondly, at least Aaron was a major Biblical character, being the first High Priest, but Imran? Imran/Amram was an extremely minor character in Scripture. We aren't given any details of his life. What's Muhammad's evidence that there was a man named Imran, who also decided to name his kids after the first Imran's, and yet all of this escaped both genealogies of Christ in the New Testament, and all of church history, until he showed up 600 years after the fact?
Muhammad got confused, and made a mistake that could only be made by someone who was vaguely familiar with the Bible could make - which fits his profile. He had, at best, an oral familiarity with a few of the major Bible stories from the Old Testament. He was not a prophet sent by God, but instead a usurper, trying to ride on the coattails of the great prophets of the Bible, which he knew next to nothing about - as demonstrated by this blunder.
Muhammad also committed a similar blunder with Saul and Gideon, as covered in this article.