Does the original N64 Zelda: OoT have small black borders like this or is it my TV accessory?

The overscan area is a lot larger than you think. CRT televisions were not precision devices. I've shaded the overscan area (technically, overscan + action-safe areas) of your screenshot in red -- that's the area that may or may not be drawn depending on the exact display used.

The black bars are part of the overscan area known as the horizontal blanking area: the period during which the electron beam is moving back to the left-hand side of the screen. Very few CRTs would show this, since it requires both an unusually large display area and a fast beam retrace.