The two featured in the post have survived because they are believed to have had famous owners. Their very survival suggests their authenticity for that reason. The older one is not all lace but has a sizeable lace insert as an apparel and is believed to have been that of St Francis of Assisi, who died in 1226. Yes, St Francis of Assisi, Il povorello, wore a lace-trimmed alb. Probably a present from an admirer.
The other one, and that is virtually made entirely of lace, is believed to have belonged to Pope Boniface VIII (1294-1303) and to date from 1298. Now Pope Boniface is much more the sort of cleric one might associate with such liturgical sartorial flamboyance - he did, after all make the Papal tiara into the triple tiara, but that’s for another post.
The illustrated article about these two remarkable survivals can be viewed at