If the owner/manger seem nice/laid back, it’s probably fine. It probably really is just so they don’t have to go through whatever corporate requires of them when an employee leaves/quits and they don’t have to onboard them just for her to pick up a shift here and there.
OTOH, if the owner is an asshole, depending on corporate guidelines and UI laws for the state, there’s probably ways for it to be used against her. I’m picturing them calling her to pick up some shifts during a busy season (ie December) and then firing her for not showing up. But if an employee isn’t going to be working, most employers would rather have them quit than have to fire them, easier all around (for the employer).
The other potential benefit for the employer is that for certain types of covid assistance, you have to keep your employment numbers up since the assistance is given to you based on you retaining employees, but I doubt that’s the case here. I’m really guessing it’s just the offboarding/onboarding.
Anyway, benefits for the employer aside, I wouldn’t worry about it. If it presents itself as a problem, just have her write a letter says “I’ve enjoyed my time working here. This is my resignation, effective immediately” and drop it off.
The only thing I’d add, is if that if you are on any kind of government assistance (state funded health insurance/food stamps/low income housing etc) or there’s financial assistance being used for college, her still being employed there may play into that. I’ve had to write notes for people previously employed by me saying that they were no longer employed because whoever is reviewing the case will consider it a source of income unless you can show you no longer work there.
TLDR, it’s fine, if it’s a problem, she can officially quit. Plus, it’s nice to know she can pick up a shift here and there if she wants. Also, she should consider it a complement. When employees leave for college, it tends to be either ‘bye Felicia’ or ‘you’ll always have a job here if you want it, if you’re home for the weekend or during Christmas/spring break/next summer, let me know if you want to work’. Your daughter is clearly the latter.