Our only source for Doria supposedly owning a copy of the image comes from a written testimonial by Italian scholar Antonio Domenico Rossi (1788-1861).
Discovered in Genoa in the palace of His Excellency the Lord Prince Doria two images of the Madonna of Guadalupe, one may have been a gift from Holy Excellence the lord Cardinal Giuseppe Doria, and it was that which with all certainty, as is recounted in the archive of that most noble family, had touched the original. It had been donated by His Catholic Majesty the King of Spain to the immortal Andrea Doria, grand admiral of the Spanish, to serve as the icon of the chapels of the principal galleys, which that celebrated captain directed; indeed it would have to be in the very same time as the celebrated battle of Lepanto, in which, by the intercession of Mary, Christendom had a most signal victory over the Turk.
Only the first part of this testimony is well substantiated (that one of the copies of Guadalupe in Genoa was donated by a Doria, Cardinal Giuseppe Doria Pamphili (1751-1816), and that this copy touched the original). The rest of Rossi’s claims is problematic, however (these are really just his personal theories).
First off, Grand admiral Andrea Doria (1466-1560) was already dead by 1571. His grandnephew, Giovanni Andrea Doria (1539-1606), did participate in Lepanto, but he was only squadron commander.
As for the actual image itself, the Doria family did have it as far back as 1684, but it doesn’t seem to have been acquired or even made during the late 16th century.
We know that around the 1670s, a member of the Doria family (Violante Lomellini Doria (1632-1708), widow of Andrea III (1628-1654) and mother of Giovanni Andrea III (1653-1737)) wanted a copy of the image. (This would have been odd if the Doria family had already possessed a Guadalupan icon of illustrious history.) A Jesuit priest went to Mexico in 1675 to get a copy for her. There, the priest found out that the native painters competed annually among themselves to see who could make the most faithful copy of the image, and so he got the copy that got first place that year. This is the image that was passed on to the Doria family.
Now the thing about the Doria image (in Aveto, Italy) is that it’s very similar to the copies of the image that were made by a late-17th century Mexican painter, Luis de Texeda (who was famous at the time for precisely that - painting copies of Our Lady of Guadalupe); in fact, the Aveto copy matches his style. If that’s the case, then the copy couldn’t have been made in the 16th century.