No, it’s not. This is actually normal behaviour for the Gameboy and Gameboy Pocket black & white series.
When you boot the device without a cartridge it will not show the Nintendo logo but instead only a scrolling black rectangle with the Registered mark next to it. Your Gameboy should be in full working order but test it with a game cartridge just to be sure.
Also, if the image is not displaying at all don’t forget to adjust the contrast of the Gameboy’s screen by spinning the Contrast wheel located on the right side of the device.