It's possible that The Fray called themselves that because when people listen to them they want to fray their skin with sharp objects, amirite?

Filet*?

No, fray is what happens if you cut fabric and don't hem it and let it drag around for a couple of weeks. Just because you cut it doesn't mean it is automatically frayed.

Fray is also what happens to fabric when you cut it.

Unless you use pinking sheers, fabric usually frays at least a little when you cut it.

It also depends on the fabric. Velvet, for example, tends to fray a little even if you do use pinking sheers.