CM_Dayton's amazingly detailed answer is spot-on, IMHO, and covers a lot of the societal aspects. As someone who reads a lot of rationalist sci-fi, though, I have to point out one inevitable subplot:
However, other than your memories and environment (e.g. you're in a mansion, so probably were doing ok...), there is no way to prove which balance was previously yours. Backup data has also been altered to be consistent. Every detail apart from the associated names is accurate. If account 123 sent money to account 456, that's still recorded, but account 123 probably belongs to someone other than expected.
This sounds like a perfect use-case for traffic analysis. All we need is a complete set of banking records and a few smart programmers, and we can start crunching the numbers and restoring the state of the world by building outward from a trusted root. Thus:
<PERSON> is the security engineering lead for Chase Manhattan. On Tuesday morning, he wakes up to the ringtone that means a critical production issue. (This is unusual, because <PERSON> is like the third person in the PagerDuty escalation. Things never escalate to him unless the site is actually down or something.)
Well, crap. Nonsense in the banking records. The automatic disaster-recovery plan was already tried, and somehow the backups are wrong too. (Everyone on the engineering team is confirming: their own balance is definitely incorrect.) Okay, check the tape archives. Somehow they're also bad. Now it's getting surreal.
90 minutes later. The site is "down for maintenance" to prevent further chaos. <PERSON> and <PERSON> (and about three junior engineers that nobody cared to forbid) are in a breakout room whiteboarding new ideas. <PERSON> is finally en route to the office, on a call with about three VPs. Nobody knows where Legal is. Someone go pull a junior engineer out of the brainstorming session and get them to make sure Legal is in PagerDuty by EOD, okay?
120 minutes. <PERSON> (who's always been a bit rogue — regulations shmegulations) has written a script to scan all the accounts and locate the ones with balances in a given range, and is going desk to desk asking if anyone remembers exactly how much money was in their account yesterday. 588
Pretty soon he has a couple of hits; some people have the app and look at their balance like the rest of us check Facebook.
There are still millions of accounts that could be theirs. This is going to be tough. But <PERSON> says she's got direct deposit on her rent — now we're cooking! (<PERSON> and <PERSON> are still holed up in the breakout room, but we can see the left half of the board is full of abstract graphs and the right-hand side is a list something like "paycheck / rent / credit card / big-ticket / ...") Twenty minutes later, <PERSON> has v2 of his script. It finds <PERSON>'s $1035.21 rent payment in a flash. One down!
In fact, two down, because we also now know the identity of <PERSON>'s landlord. (Well, somewhat. His name is <PERSON> something? He lives down in Gilroy. Never seen him. But we can get him on the phone if we need to.)
<PERSON> uses his script on <PERSON>'s direct-deposit paystub. (Normally <PERSON> would have an aversion to revealing his salary, but today isn't "normal". Normally <PERSON> would love to reveal everyone's salary. He's having a ball right now.) Now we have <PERSON> and <PERSON>. This is going great! And <PERSON> says he paid <PERSON> for lunch the other day, it was like $20, so if they can see that payment they'll know which account is <PERSON>'s... oh. Hmm. <PERSON> paid <PERSON> with Venmo. So it happened entirely within the Venmo app, and won't be reflected in anyone's Chase balance. Bother. Venmo — and credit cards in general — are going to make this a lot harder than if everyone paid by check all the time!
Here I'm out of my depth and so I'll stop with the details, but hopefully <PERSON> and <PERSON> will come up with something. <PERSON> has some contacts at Netflix; maybe their billing system is still unscrambled, in which case we can resolve a lot of accounts by looking at the timing of their monthly $7.99 payments. (Or whatever it is now. <PERSON> does not have the Chase app. 588