It looks like it could be a hotel (the sign above the note looks like it might be a room number/front desk placard) so maybe they also have issues with fake id holders also using stolen credit cards or just not paying the bill at all.

Because if it can be proven in court that the ID was identifiably fake, they can be held liable for any damages should the minor become intoxicated. "Checking" an ID means there was a reasonable attempt to verify the legitimacy of the document and that it matches the individual using it.

Since what is identifiably fake is a somewhat vague category that can be litigated in court, most bar and booze store owners try to err on the side of too careful--which usually just leads to annoying people--rather than not careful enough, which can lead to fines, damages, or the loss of license to serve alcohol. Any of those can be bad enough to close a business.

Also to add, this looks like a doctor's office/pharmacy and not a liquor store. Probably trying to keep people from stealing shit or getting prescriptions under fake names.

It's easier to just say no and not take the business if you think it's a risk. Don't wanna lose your liquor license to a babyfaced undercover cop testing fake IDs. I'm not saying our laws on alcohol aren't ridiculous, but as a business owner it makes sense to protect your ability to sell alcohol when it's your money maker.