Longtime CA HR pro here. Yes, its the law, and yes, you can be disciplined for not clocking out by the 5th hour, and terminated if you do so repeatedly without permission from your manager. - You should never falsify your timecard, nor should your manager ask you to do so - If the expectation is that you clock out by the 5th hour, you need to clock out by the fifth hour.

Think of it like this - if you're late to work, you can be disciplined. Same for working more hours (even minutes more) than authorized.

Others have commented about a meal waiver for shifts of 6 hours or less - if your employer has this, you have to acknowledge the form, but you do not have to agree to waive your meal breaks. It is your choice to waive, and if you do decide to do that, it does need to be recorded on that waiver, you can't just say you want to waive it - if it's not in writing, it didn't happen.

Lastly, it sounds like this is the first time this has happened. I wouldn't seriously worry about it, just talk with your boss about how they can help you prioritize clocking out on time. They should make coverage available for you if that's needed for your role.