A person they don't know that seems familiar.
"Where did this picture come from... seems familiar. Gods, when was this? ... Eh, probably not important."If you've ever been to a family reunion, break out a photo album of the occasion and look around. You'll get that reaction eventually."That's a familiar name... do I know them? Gods, I can't remember... probably not."
Yes.
Possibly, they'd remember that someone they cared about went into the battle, but not remember who or what happened to them - simply accepting that this is common among people now.
Possibly, they'd think someone was messing with them by leaving pictures of strangers up everywhere. What if you went to the kitchen five minutes from now and there was a picture of a baby with the words, "My Daughter" in your handwriting under it stuck to the fridge. Would your first thought be, "Wow, I must have a daughter I forgot about!" or "WHO IS F$#!ING WITH ME? SHOW YOURSELF."
Etc. Etc. Etc.
Have you ever seen people with real amnesia? It gets pretty bizarre. For instance, Clive Wearing could not form new memories and could only stay lucid in given moments. Without the ability to connect one moment to the next, he'd keep a journal and write over and over, "For the first time, I am truly, completely awake." He wouldn't recognize the past entries - they were in his hand writing, they expressed similar sentiments, but according to everything about his experience it wasn't him. At times it would make him furious and he'd scribble out the past entries that were lying to him. Even Alzheimer's can make you treat the things you once held dearest as completely unfamiliar.
My point is that there's little way to look at this logically. Issues that interfere with mental functioning interfere with the logical tool and break every sensible standard you can think of. Read up on Clive Wearing, Henry Molaison, Phineas Gage, Patient KC, etc. and you'll get an appreciation for how easily you can lose your "soul" long before you're dead. What's happening in Eorzea is just specific and we're meant to assume there's a specific cause. Mental phenomena manifesting via magickal acts, most likely; not so strange, fictionally speaking.