What effect does the patron deity and name day have?

[Enkidoh]

Also I think it might make an appearance during the Ceremony of Bonding. It's been a while since I actually got married, but I'm sure the symbol of the two dieties of the bonding partners show up.

Correct - the altar inside the Sanctum of the Twelve shows the symbol of your character and your partner's guardian deities during the ceremony (one one either side), but apart from that, the only other benefit your chosen guardian used to grant was a slight native elemental resistance to that particular element (for instance, my character's guardian, Azeyma, granted a slight resistance to fire), although that was abolished when Stormblood launched.

Back in the early 1.0 era you could also evoke your guardian during levequests in the form of a buff called 'Guardian's Blessing' which increased the EXP you gained, but that was replaced by rested EXP and Guardian's Blessing was removed.