Can "The Virgin Mary" be called "daughter of Aaron"?

Probably not.

The precise relationship between Mary & Elisabeth is not known. συγγενίς could be a cousin, but it just means relative.

• Elisabeth was a descendent of Aaron (and thus of the tribe of Levi) on her father's side (Luke 1:5).
• It appears very likely that Mary was a descendent of David (and thus of the tribe of Judah) on her father's side. If either Matthew or Luke is providing Mary's genealogy--a view that has been common among Christians since the Ante-Nicene era--then Mary is definitely a descendant of David on her father's side.

If Mary & Elisabeth were first cousins, then one of these two possibilities would be true:

• Elisabeth's mother was of the tribe of Judah
• Mary's mother was of the tribe of Levi

If Mary & Elisabeth were more distantly related than first cousins, quite a few additional permutations are possible.

So, Mary's mother may have been a "daughter of Aaron", making Mary a matrilineal descendent of Aaron, but this is not certain. Even if it were the case, Jewish genealogies & tribe membership were traced patrilineally, so Mary would not be considered a "daughter of Aaron" in the sense that Elisabeth was.

Can "The Virgin Mary" be called "daughter of Aaron"?

To a certain degree, according to our present knowledge and understanding of the genealogies of Jesus and Mary, it might be possible to call Mary, the daughter of Aaron. There remains a nuance to be understood here and it is not at all absolutely clear at present.

Genealogy

The New Testament tells little of Mary's early history. The Gospel of Matthew does give a genealogy for Jesus by his father's paternal line, only identifying Mary as the wife of Joseph. John 19:25 states that Mary had a sister; semantically it is unclear if this sister is the same as Mary of Clopas, or if she is left unnamed. Jerome identifies Mary of Clopas as the sister of Mary, mother of Jesus. According to the early 2nd century historian Hegesippus, Mary of Clopas was likely Mary's sister-in-law, understanding Clopas (Cleophas) to have been Joseph's brother.

According to the writer of Luke, Mary was a relative of Elizabeth, wife of the priest Zechariah of the priestly division of Abijah, who was herself part of the lineage of Aaron and so of the Tribe of Levi.[65] Some of those who believe that the relationship with Elizabeth was on the maternal side, believe that Mary, like Joseph, was of the royal Davidic line and so of the Tribe of Judah, and that the genealogy of Jesus presented in Luke 3 from Nathan, is in fact the genealogy of Mary, while the genealogy from Solomon given in Matthew 1 is that of Joseph. (Aaron's wife Elisheba was of the tribe of Judah, so all their descendants are from both Levi and Judah.) - Mary, mother of Jesus

Although the Quran calls Mary "the daughter of Imran" and it mentions that people called her a "sister of Aaron, it is not something typically invoked by Christian communities. Nevertheless, the Christian understanding of Mary’s genealogy and her role in the Church is quite different in relationship to the understanding of Mary in Islam.

Prat, S.J. & Heenan, S.J., Jesus Christ: His Life, His Teaching, and His Work vol. 1, p. 62:

What relationship existed between Elizabeth, daughter of Aaron and of the tribe of Levi, and Mary, of the line of David and the tribe of Juda? Their difference in age suggests that Elizabeth was Mary’s maternal aunt. It may be, on the other hand, that their mothers were sisters. That is possible, too, because there was nothing against a woman of the house of Aaron marrying a member of the tribe of Juda. But whether Mary was the niece or the cousin of Elizabeth, they were certainly close relatives, which is all that St. Luke says and all that we need to know.

OP: could the Jews of the first century, have called Mary "the daughter of Aaron", even if she was only akin to Elisabeth by the tribe of Judah by her mother's side?

Mary was of the tribe of Judah, of David's seed.

Concerning his Son Jesus Christ our Lord, which was made of the seed of David according to the flesh; Rom 1:3

The word translated by KJV as "cousin" at Luke 1:36 is "syngenes", which may also mean kin or of the same nation. Various other bible translations use it in that wider sense.

RSV And behold, your kinswoman Elizabeth in her old age has also conceived a son; and this is the sixth month with her who was called barren.

YLT and lo, Elisabeth, thy kinswoman, she also hath conceived a son in her old age, and this is the sixth month to her who was called barren;

Additionally, there is the same sense of specifically not being of Levi in Hebrews.

For it is evident that our Lord sprang out of Juda; of which tribe Moses spake nothing concerning priesthood. Heb 7:14

Further, to add, scripture calls for Messiah to be of the line of David, not Aaron.

Hath not the scripture said, That Christ cometh of the seed of David, and out of the town of Bethlehem, where David was? John 7:42

The LORD hath sworn in truth unto David; he will not turn from it; Of the fruit of thy body will I set upon thy throne. Psalm 132:11

And Jesus answered and said, while he taught in the temple, How say the scribes that Christ is the Son of David? Mk 12:35

So, to answer the OP, no, it would be impossible for Mary to be of Levi, rather than Judah.