I am going to guess, the girl has a split personality she developed. The replica is just the person she wants to be, and doesn't realize when she switches personalities.
Sunao is not a shut-in. She go to school and she got friends too. So it's not like she shut herself in her room everyday.
She only summon Nao on days when she doesn't want to go, for example when there's test, tiring sport (such as long distance running), or most regularly when she's having her period (she seems to be the type who had it terrible, lots of pain and headache).
About the mechanism...
When erased, the replica just go to sleep mode, so it's not like they're shut in void and despairing. The next time they know is they're summoned again.
Replica has access to the original's memory at the point they're summoned, although it very depend on how much the original can recall it (so things like what you ate yesterday is much harder to recall). They see it like "reading a story". This doesn't work both way though, so original can't sync the memory from replica.
Also... big spoiler
No, Replica can't live independently from Original. Even if a Replica is destroyed, Original can summon them back no matter how many times, keeping the same memory they had. But if Original die, Replica will disappear.
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This part got skipped in the manga...
Nao stored her money in cookie can. The cans are from theme park. Sunao doesn't need it anymore, but Nao got fond memories of it, so she secretly hide it, along with the 50 Yen she got from helping mom for years, inside the storage room.
It always felt heart wrenching for me when Nao count her money everytime she spend it.
Those 198k Yen is her whole wealth. With it, she feels like she can do anything, and this is finally the time she can enjoy the fruit of her dilligence.
But unlike Sunao, she doesn't get pocket money, and not like she can find part time job. So everytime she mention the money she had left, it always sounds like countdown of the time she had left. T__T