Is Earth's moon the only one where a total eclipse of the sun covers the entire sun? Is Earth's moon the only one where a total eclipse of the sun covers the entire sun?

Are any other moons the same size as the sun as viewed from their planet like Earth's moon?
As planets get farther from the Sun, the Sun takes up a smaller part of the sky.  The Sun is about 31 arc-minutes when viewed from Earth, but just 6-7 from Jupiter and 3-4 from Saturn.   Less than 2 from Uranus and about 1 arc-minute from Neptune, not much bigger than Venus appears from Earth when Venus is visibly large in the sky and when Venus transits the Sun, viewed from Earth, its size is about 1 arc-minute, about the same size that the Sun appears from Neptune.  To get a sense of what size the Sun is from Neptune, look at a photo of a Venus transit.

Jupiter has 5 satellites capable of creating a total eclipse, the 4 Galilean satellites and the tiny but close Almathea, which is irregularly shaped, so it wouldn't be the same size as the Sun.

Saturn has 6. All are larger than the Sun when viewed from Saturn.   

Uranus, because the Sun is quite small that far away, has 12 moons that can create a total eclipse.  All 12 are considerably larger than the Sun in the Uranus' sky but because of Uranus's nearly sideways axial tilt, solar eclipses are rare and only happen at the midpoints in its orbit every 42 years.

Neptune has 7, but due to its axial tilt and Triton's off-equator orbit, eclipses are rare.

Io probably casts the largest shadow.   Though Pluto and Charon cast a shadow over the largest percentage of the other object, about once every 120 years.   Two of Pluto's smaller moons, Nix and Hydra are large enough to block the Sun completely though eclipses from them are probably rare.

As far as I can tell, having checked, our Moon appears unique in our solar-system in being a nearly perfect fit over the sun.

Further reading if interested, though there's some variation in the answers, they only mention 4 of Jupiter's moons.