Is the magma in one volcano different from the magma in every other volcano?
In keeping with your Lord of the Rings inspiration, the first reference is from New Zealand, where the Lord of the Rings movies were made.

There are three main types of magma that volcanoes currently produce: basalt, andesite and rhyolite.

Basalt magma has a high temperature, around 1200ºC, it is poor in silica, has a low viscosity and a low gas content. When basalt erupts it produces a flowing magma.

Andesite magma has a mid range temperature, between 800ºC and 1000ºC, it also has a mid range silica content and moderate viscosity and gas content. When andesite erupts it flows but it is also moderately explosive.

Rhyolite magma has a low temperature, between 750ºC and 850ºC, it is silica rich and it has a very high viscosity and high gas content. When it erupts it is very explosive.

Other forms of magma which are much rarer are carbonatite melts and komatiite melts. Carbonatite melts may be cool, as low as 600ºC, whereas komatiite may have been as hot as 1600ºC. 

Unlike the other forms of magma that are composed of silicate minerals, carbonate melts are composed of carbonates and can be confused with marble and may have to be geochemically analysed.

Most komatiites are very old, up to  4 billion years old. It is speculated that komatiite magma formation was possible when the Mantle was at least 500ºC hotter.