The verse in question is;
"O sister of Aaron! Thy father was not a man of evil, nor thy mother a woman unchaste!" (Al-Qur’an 19:28)
It does not mean that here Qur’an calls Mary, the real sister of Aaron, the Prophet who was hundreds of years before her. Here she is only being called a female person of the family, from the Noble lineage of Prophet Aaron.
The Qur’anic usage:
In order to understand Qur’anic language one must look into other Qur’anic passages.
Qur’ans calls Prophet Shoaib as the brother of the people of Midian. Qur’an says;
“And unto Midian (We sent) their brother, Shu'eyb." (Al-Qur’an 7:85)
And similarly Qur’an calls Prophet Salih, the brother of the people of Thamud. It says;
"And unto Thamud (We sent) their brother Salih." (Al-Qur’an 11:61)
In both these examples it is never meant that the Prophets were the real brothers of the each and every person of that tribe. It’s only way to address them. It means that Shu'aib was a (male) person from the people of Midian and similarly Salih from the people of Thamud.
So in the very same manner when Qur’an describes Mary, the mother of Christ as 'Sister of Aaron', It means that she is being referred to as a (female) person from the people, the lineage of Prophet Aaron.
This is infact an Arabic idiom, a way to address. In Arabia a person from the tribe Banu Mudhar may be addressed as Ya Akha Mudhar, meaning 'O the brother of Mudhar'.
The Prophetic answer to this question:
Mughira b. Shu'ba reported: When I came to Najran, they (the Christians of Najran) asked me: You read" O sister of Aaron", whereas Moses was born much before Jesus. When I came back to Allah's Messenger (may peace be upon him) I asked him about that, whereupon he said: “The (people of the old age) used to call names (of their persons) after the names of the Prophets and pious persons who had gone before them.” (Sahih Muslim, Book on General Behaviour, Hadith 3962)
The Holy Prophet (PBUH) meant to say that Mary’s relationship with Prophet Aaron was mentioned because she was of his lineage for among Semites it was common to associate a person with his lineage.
This fits the context as well. The people thought that Mary had done something unworthy of a chaste woman and thus as they rebuked her, they first made a reference to her noble ancestry and then testified that even her own father was no wicked person (verse 28).
Observations from the Bible:
1- One thing we need to consider is that Qur’an does not call Mary ‘Sister of Moses’ but ‘Sister of Aaron’.
2- Aaron (PBUH) we know was the first in line for the Israelite priesthood.
3- New Testament clearly tells us that Elizabeth, the mother of John the Baptist (PBUH) was from the lineage of Aaron (Luke 1:5). Also it tells that Mary was a cousin of Elizabeth (1:36). Therefore we can easily call them both ‘Sisters of Aaron’ i.e. from the lineage of Aaron.
4- Infact we have evidence of the similar usage from the Bible. In Genesis 13:8 Abraham and Lot are called brothers while certainly they did not come from loins of one individual. It reads;
“And Abram said unto Lot, Let there be no strife, I pray thee, between me and thee, and between my herdmen and thy herdmen; for we be brethren.”
A certain Christian commentator, Adam Clarke says in his commentary to this verse;
‘[It means] we are of the same family, worship the same God in the same way, have the same promises, and look for the same end.’
Was the same not true in case of Mary and Aaron? Were they both not from the same priestly family and of same faith?
So the conclusion is that this is only a beautiful way of expression in the Arabic language and to try to twist it some other way speaks only of ones inner wickedness and that’s all!
INDEED ALLAH KNOWS THE BEST!