Is Zion associated with Mary in Catholic doctrine? In the accepted answer to this question, How is the Catholic interpretation of the Woman of the Apocalypse as being Mary reconciled with the dogma of perpetual virginity?, I find reference to Isaiah 66:7:

“Before she was in labor she gave birth; before her pain came upon her she delivered a son. 

This is given as a prophetic foretelling of the Catholic belief that Mary did not suffer pains (nor was her hymen even ruptured) in birthing the Christ child.  Reading on in the Isaiah passage it becomes evident that the &quot;she&quot; who is in labor and delivered a son prior to the pain is Zion:

Who has heard such a thing? Who has seen such things? Shall a land be born in one day? Shall a nation be brought forth in one moment? For as soon as Zion was in labor she brought forth her children. - Isaiah 66:8

Zion (Tsiyon in Hebrew) was a Canaanite hill fortress which was captured by King David and thereafter referred to as the &quot;City of David&quot;.  It can be understood Scripturally as the most ancient areas of that hill, Jerusalem as a whole or, spiritually, as the Kingdom of God.
In order for this passage to foretell Mary's painless childbirth there would have to be some connection between Zion and Mary.  Is there a teaching in Catholicism that equates Zion with Mary in some fashion?  If so, does this have any biblical basis at all or is it pure tradition?
In the Litany of Loreto, which apparently was composed to suppress other Marian litanies. I don't know if they were extravagant or borderline heretical or what.  But that's what the intention was so there must have been something behind it.

The place of honor it now holds in the life of the Church is due to its faithful use at the shrine of the Holy House at Loreto. It was definitively approved by Pope Sixtus V in 1587, and all other Marian litanies were suppressed, at least for public use. Since then, several Popes have added titles to the Litany, as only they may do.

But, what the Litany does leave in is the reference to Mary as the &quot;Tower of David&quot;, which is a reference to Song of Songs.

&quot;Thy neck is like the Tower of David built with turrets, whereon there hang a thousand shields, all the armor of the mighty men&quot;

History seems too confusing to me to place the Tower of David on Mt. Zion, but it's clear that Medieval Christians thought it was on Mt. Zion and I'm not sure why it would be a long way away from Jerusalem.
The other Marian connections to Jerusalem, the Bride of Christ, etc... are pretty firmly entrenched in the minds and traditions of all Christians except those descended from Protestant sects today.  They certainly had a hold over the popular conception of the Virgin Mary in the Middle Ages, but it's always been up in the air whether those ideas came from just Vox Populii or Vox Dei or a combination of the two.
Suffice it to say, Constantine's mother St. Helena was enthralled enough to try to bring Our Lady's house to Italy, it hit a few bumps along the way, but miraculously reappearing houses are wont to do that and Catholics don't put limits on God's works or question His designs. That's there the Loreto part comes in, as the Litany mentioned above is the Litany of Loreto and the house itself, found its final resting place in the same town.
So 1000 years later, 1000 years of accruing traditions and codifying doctrine we wind up with a popular devotion that still contains a veiled reference to Mary and Zion as one that made the cut.

HOWEVER, and to the contrary, from the passage of Isaiah that you quoth. I was reading through the Little Office of Our Lady which seemingly contains every veiled and unveiled reference to Mary in the Old and New Testaments and did not come across that particular passage.  I expected it to be there and it wasn't so it might be the case that nobody has prescribed any particular typography of Mary to this scripture.