As the title says: A coworker and I had recently made a mistake at work. In response to this, our manager decides we have to work for two extra hours on our next shifts unpaid. Is this legal? I work in the State of Georgia in the US, by the way.
This is illegal, under the Fair Labor Standards Act, assuming you are not an "exempt" employee, which is primarily a salaried employee (your pay isn't based on how many hours you work). There are details about the complaint process here. It is useful to know that retaliation against an employee filing a legal complaint is also illegal.
Per the article "Overtime Rules for Exempt and Non-Exempt Employees":
The Department of Labor (DOL) has rules for when employers must pay overtime to employees. The DOL assumes every worker must receive overtime pay if they work over 40 hours in a week, at a rate equal to 1.5 times their hourly rate (at a minimum).1 But some employees, because of the nature of their work, are considered to be "exempt" from overtime pay.
It used to be that the terms "exempt" and "non-exempt" were clearly defined. But the DOL has more rules to protect lower-paid exempt employees from falling below the minimum wage, by requiring that they must be paid overtime."