These virgins need a disambiguation.
On the one hand, La Virgen, Is Our Lady, Blessed Mother. She's the only Holy Mother of God the Blessed Virgin Mary. That's her with her place of honor on the side of the altar at most Catholic Churches throughout the world. If you hear of her under the title of La Guadalupana (Our Lady of Guadalupe), Our Lady of Lourdes, Our Lady of the Snows, Our Lady of Fatima, Our Lady of La Salette, Our Lady of Kibeho or any of the titles in the Litany of Loreto then that's still her. It's a worldwide thing, but Latin America is particularly fond of Our Lady of Guadalupe, and with good reason, according to this book her apparition in Mexico is responsible for millions of conversions and the end of human sacrifice!
We don't have her remains since she was assumed into heaven, so the statues are just statues. Every church has an altar with some relic of a saint in it (anything from a bone to something a saint wore or a piece of their casket). Sometimes relics are transported from place to place in a procession (which is a lot of fun). It wouldn't be a relic of Our Lady though. Statues are blessed and are sacramentals, so they do have some significance in the Church, but aren't to be prayed to as idols ever. There probably is a bit of pride in having a beautiful statue of La Virgen, but that's about it.
The titles that Mary goes by more akin to those given to her Queenship, like all the titles you'd see the King of France go by (and go through). They're not different virgins, they're titles she is given (by the grace of God, as even the king of France would have said) when she conquers a nation for her Son (in a real an spiritual sense).
On the other hand you have the holy virgins. Unless otherwise noted, those are the female saints who took vows be a spouse of Christ who didn't die a martyrs death.
Male saints who take similar vows are usually referred to as Confessors, not because they heard a lot of confessions, but because they confessed the faith. Male chastity is every bit as reverenced and useful as female in the Church. Can't give you a legit explanation of why men aren't titled under 'virgin', but keep in mind most church words (Ordinary, Common etc...) have completely different connotative meanings than in ordinary and common usage. (that was a Catholic joke, har har).
Regarding the title 'virgin'
Catholic calendars usually say, St. Agnes Virgin, Martyr to signify which prayers are supposed to be read in the Liturgy of the Hours (unless otherwise noted the section you read for a saint like St. Agnes is called the 'Common of Virgins') it also signifies what colors are supposed to be worn by the priest at Mass. Red for martyrs, white for other saints (including virgins).