Luke was Greek and they have a word for cousin.
I believe the word in question is "
adelphos " ( although I think we’ll find the passage(s) in reference to Jesus’ brothers is found in the Gospels of Matthew and Mark - but not Luke).
. . . The Church also teaches that calling them a brother or a sister was the only way to describe them as cousins. . . [Luke 1:36]
Could you please supply a link/quote for the one assertion quoted above which it is claimed the Church teaches - so we could see it in context ?
There’s a plausible explanation of Elizabeth’s relation to our Blessed Mother which includes even a flowchart with its reasoning at this link :
defendingthebride.com/ma2/brothers.html
There is a lot of other material available both here at CAF and elsewhere giving proofs regarding the question of any possible biological siblings of Jesus. They also expound upon the word
adelphos and its connotations.
Some such sites ask the logical question that that if Mary had other children biologically then it would’ve been their duty to take her into their care when Christ died ; so and why didn’t they and why weren’t they there at the foot of the Cross ?
I’m having a difficult time understanding the OP’s phrasing . . .

. . .
If the ultimate question is
. . . if this is true then why is Elizabeth called “cousin” in the Bible.
. . . then why is the thread entitled
Mary’s Virginity ? The fact of whether Elizabeth was a biological cousin of Mary or not has absolutely nothing to do with our Blessed Mother’s virginity.
So the rationale here, as far as I can tell (correct me if I’m wrong) appears to be:
If the Archangel Gabriel , according to the Gospel of Luke ,calls Elizabeth Mary’s
cousin ( we note in the same chapter of Luke’s Gospel, describing the same event of the Annunciation that it is also specified twice that Mary was a
virgin [Luke 1:27] ) then this is somehow supposed to call into question Mary’s virginity.
Sorry , I can’t really follow the reasoning on this one, but at any rate, here’s another good article to go along with bzkoss236’s links - this one from EWTN, by someone who did the homework on the etymology and which should help clear up the confusion ; wherever it lies :
Brothers and Sisters of Jesus
(* Everybody say “ah”

)