I think the Apostolic Blessing in this case would attach to the persons saying the DM, not to the persons being prayed for. The idea is if I say the DM before the Blessed Sacrament for “Jane”, who is suffering and dying in the next hour, then Jane gets the benefit of the prayers and I get the Apostolic Blessing, which might carry some indulgence or merit that I could then give to a soul in Purgatory.
I have no problem thinking Pope JPII gave all the members of this lay apostolate his Apostolic Blessing on some particular day as a one-time thing when he was alive. Popes do that sort of thing all the time.
I am however dubious that everybody praying the DM in Adoration gets an Apostolic Blessing for all time or each time they say the DM in Adoration, including presumably long after the Pope who made the grant is deceased.