Yes! The delightful reason why this configuration can repeat within itself is that this Game of Life pattern is a machine with the rules of Game of Life built into it. It's calculating and displaying the game step-by-step. If you watch the machinery you can see the "data" flying around between cells and then the cells turning on and off.

This pattern isn't unique – you could certainly come up with infinitely many more machines in Game of Life that do the same thing. But yes, most Game of Life patterns aren't fractal like this.