As I recently mentioned, I bought a book for the Triduum about the corporal and spiritual works of mercy. The author mentions at one point:
Prior to opening the [second Vatican] Council, Pope John XXIII made a pilgrimage to Assisi, where he placed the Council under Saint Francis’ special patronage and prayed that he who was called “the father of the poor” in his own time would intercede for the Church so she would recognize herself once again as a Church “of the poor and for the poor.”
This seems to be where Pope Francis got his tag-line. Anyhow, since JPII’s pontificate was all about enacting the Council, and BXVI’s pontificate was all about correcting wrong understandings of the Council, it seems to me no surprise that Pope Francis’s name and schtick would have do with the intention behind Vatican II.