The Quranic verse 19:28 is:
يَا أُخْتَ هَارُونَ مَا كَانَ أَبُوكِ امْرَأَ سَوْءٍ وَمَا كَانَتْ أُمُّكِ بَغِيًّا
Ya okhta haroona ma kana abooki imraa sawin wama kanat ommuki baghiyyan
The Arabic word used here is similar to the Hewbrew word אָחוֹת (achowth), its usage is explained in Gesenius' Hebrew-Chaldee Lexicon as:
Source: Blue Letter Bible, scan on archive.org
Its similar to the Arabic Akhi (brother) which also has cognates in Hebrew where they include tribal relations and similarity of character:
Source: Blue Letter Bible, Scan on archive.org
From the NT, there seems to be evidence that Mary is related to the descendants of Aaron, brother of Moses, and it could be that the people referred to that:
Luke 1:5 There was in the days of Herod, the king of Judaea, a
certain priest named Zacharias, of the course of Abia: and his wife
was of the daughters of Aaron, and her name was Elisabeth.
Luke 1:35 And the angel answered and said unto her, The Holy
Ghost shall come upon thee, and the power of the Highest shall
overshadow thee: therefore also that holy thing which shall be born of
thee shall be called the Son of God.
Luke 1:36 And, behold, thy cousin Elisabeth, she hath also
conceived a son in her old age: and this is the sixth month with her,
who was called barren.
I understand that this is a valid interpretation among Christian scholars, though some may disagree with it:
From Elliot's commentary to the Bible for English readers:
Taking the word in its usual sense, it would imply that either the
father or the mother of Mary had been of the house of Aaron, or that
the mother of Elizabeth had been of the house of David.
The scanty notices in the Gospels are (1) that she was a “cousin,” or
more generally a “kinswoman,” of Elizabeth, and may, therefore, have
been, by her parentage, wholly or in part of the daughters of Aaron.
From Expositor's Greek Testament:
Luke 1:36. καὶ ἰδού, introducing a reference to Elizabeth’s case to
help Mary’s faith.—συγγενίς, late form for συγγενής (T.R.), a blood
relation, but of what degree not indicated, suggesting that Mary
perhaps belonged to the tribe of Levi.—γήρει: Ionic form of dative for
γήρᾳ (T.R.). Hellenistic Greek was an eclectic language, drawing from
all dialects as from the poets, turning their poetic expressions to
the uses of prose.—καλουμένη: Elizabeth is described as one who is
still being called barren, though six months gone in pregnancy,
because people have had no means of knowing her state.
From Meyer's NT Commentary:
συγγενίς] The nature of this relationship, which is not at variance
with John 1:36, although questioned by Schleiermacher and others, is
wholly unknown. It is, however, possible that Mary was of the stock of
Levi (so Faustus the Manichean in Augustine, c. Faust. xxiii. 9; and
recently, Schleiermacher, Schr. d. Luk. p. 26; Hilgenfeld, Ewald,
Gesch. Chr. p. 177, and others), as the Test. XII. Patr. p. 542 makes
the Messiah proceed from the stock of Judah (Joseph) and (comp. p.
546) from the stock of Levi.
Secondly, the Quran contains an account of the life of Mary before Jesus's birth and states that she was dedicated to the Temple and raised under guardianship of the Kohen:
Quran 3:35 [Mention, O Muhammad], when the wife of 'Imran said,
"My Lord, indeed I have pledged to You what is in my womb, consecrated
[for Your service], so accept this from me. Indeed, You are the
Hearing, the Knowing."
Quran 3:36 But when she delivered her, she said, "My Lord, I
have delivered a female." And Allah was most knowing of what she
delivered, "And the male is not like the female. And I have named her
Mary, and I seek refuge for her in You and [for] her descendants from
Satan, the expelled [from the mercy of Allah ]."
Quran 3:37 So her Lord accepted her with good acceptance and
caused her to grow in a good manner and put her in the care of
Zechariah. Every time Zechariah entered upon her in the prayer
chamber, he found with her provision. He said, "O Mary, from where is
this [coming] to you?" She said, "It is from Allah . Indeed, Allah
provides for whom He wills without account."
Although this account is not present in the canonical books of the New Testament (as far as I am aware), its referred to in Catholic and Eastern Orthodox tradition and in some books of the apocrypha.
Wikipedia on the Presentation of Mary:
The account of the Presentation of the Blessed Virgin Mary in the
Temple is principally based on the Protoevangelium of James, which has
been dated by historians prior to the year 200 AD. The story relates
that in thanksgiving for the birth of their daughter, Mary, Joachim
and Anne decide to consecrate her to God, and bring her, at the age of
three years, to the temple in Jerusalem. Mary’s presentation in the
temple draws parallels to that of the prophet Samuel, whose mother
Hannah, like Anne was also thought to be barren, and who offered her
child as a gift to God at Shiloh.
Mary remained in the Temple until her twelfth year, at which point
she was assigned to Joseph as guardian. According to Coptic tradition,
her father Joachim died when Mary was six years old and her mother
when Mary was eight. While the story is a legend with no foundation
in history, the point is to show that even in her childhood Mary was
completely dedicated to God. It is from this account that arose the
feast of Mary's Presentation.
So it could be that the people referred to her as a sister of Aaron because of her link to the Priestly order, and her dedication and worship in the Temple, a similar disposition to Aaron.
Tafsir Al Qurtubi narrates the following views present in Islamic tradition on the relation between Aaron and Mary:
فقيل: هو هارون أخو موسى؛ والمراد من كنا نظنها مثل هارون في العبادة
تأتي بمثل هذا. قيل: على هذا كانت مريم من ولد هارون أخي موسى فنسبت إليه
بالأخوة لأنها من ولده؛ كما يقال للتميمي: يا أخا تميم، وللعربي يا أخا
العرب. وقيل: كان لها أخ من أبيها اسمه هارون؛ لأن هذا الاسم كان كثيراً
في بني إسرائيل تبركاً باسم هارون أخي موسى، وكان أمثل رجل في بني
Some said, the Aaron here is Aaron, the brother of Moses. The meaning
is that "we took her worship and service (to the Temple) as similar to
Aaron, then how could she do that?" Some said that she was called
sister of Aaron because she was of his descendants ... Some said that
Mary had a (step) brother from a father who was named Aaron, and that
the name was common among the tribes of Israel because of Aaron's
piety, and this other Aaron was also well known and exemplary devout person of the Children of Israel.