They're asking you to sign some legal documentation. If you forge a signature rather than getting it signed by your employer, that's fraud. If you create a false identity in order to hide the fact that you have an employer, that's also fraud.
You have exactly two legal (and, just as importantly, honest) options in this situation: Get permission from your employer before contributing, or don't contribute. If you like the project and don't want to create trouble for its maintainers, you will pick one of those two options.
In those cases, your employer would own your contributions and thus you need permission from them to license their copyrighted work (your changes) under whatever the project license is.
But in any case, contributing under a pseudonym is something that you should think about very seriously. This has been done before in the Linux kernel and luckily nobody got sued over it, but it is basically copyright infringement mixed with various levels of fraud and deception. Don't do this to us poor maintainers.
How does your employer own what you do in your free time? AFAIK there is no job contract like that which is legally enforceable. At least in California.
But I assumed that GP was talking about wanting to contribute something they did on work time, not on their own time.
Worrying about these details is the last thing a project maintainer needs to be worrying about. Easier to just require everyone to have a belt, even the ones who say they own suspenders.
Did it cause you some problems later, or?
If that permission isn't secured, though, and the employee has a contract with their employer that signs ownership of some or all of their off-hours work over to their employer, and the employer decides to try and exercise those rights, then it's anyone's guess who the real owner would be. Might vary by jurisdiction. Might be down to whether the open source project can afford to lawyer up in the first place.
Given all that, a FOSS project isn't unwise for asking for a permission slip. You could argue that it's being over-cautious, but that's the project maintainer's decision, and it deserves to be respected.
No, it's not fraud because there's no intention to gain illegal or unlawful gain. It's just deceit.
Easier said then done.