Fiber can certainly suffer from the exact same issues as copper - a neighborhood or group of homes or town or whatever connects to the Internet over a single piece of cable or logical connection, or a single piece of equipment, and those can become congested.
In theory, that could limit your maximum available bandwidth by limiting everyone's bandwidth.
But with Fiber, the infrastructure is newer and generally installed with future capacity in mind. Also, fiber is cheap and labour is expensive, so the fiber from your home may continue for some distance before it reaches the first piece of powered equipment, which helps make a physical fiber network more economical, and more redundant.
I would think a grounded box would be much less safe here. In plastic if a wire comes loose there are no repercussions