From what I've studied military slavery (at least the kind similar to that of the Mamluk Sultanate) was largely limited to the Islamic world, although that world at times encompassed areas from India to Africa.

Military slavery became popular in the Muslim world for a number of reasons. First of all was the nature of slavery in general. Unlike, say Antebellum Southern American, few slaves in the Islamic world were involved in manual labour. Most had roles in their master's household. While initially modest these roles eventually grew to encompass military and administrative duties. This meant that these slaves had a generally higher social position than slaves in most other cultures. On top of that, in theory (worth stressing that bit) the relationship between a slave and their master was supposed to be similar to that between a junior and senior partner, with expectations and responsibilities on both sides.

While Mamluks are the most famous example of Islamic slave-soldiers (and the one I have studied the most) they were far from the first. Elite slave-soldiers (usually called Ghulam) first came to positions of prominence during the Abbasid Dynasty in the 9th century. Perhaps the primary use of these soldiers (and those to follow) was as a balance against the ruler's rivals for power. Islamic armies traditionally consisted of tribal warriors loyal to their chieftain/lord, however Ghulam, who were often recruited from afar and trained from a young age, owed their loyalty only to their master.

Ghulam continued to play an important role in Islamic armies/states throughout the following centuries (see David Nicolle's Saladin and the Saracens for their role in the Crusades) and finally reached the pinnacle of their importance When Baibars overthrew the Abbasid dynasty after the Battle of Ain Jalut.

For more information (mostly on the Mamluks) see pretty much anything by David Ayalon or 'Mongols and Mamluks' by Amitai-Preiss.

Are there other examples than in the Muslim world?

One non-Muslim example would be the Chikunda employed by Portuguese landowners in Mozambique. Private Portuguese citizens had claimed vast estates along the Zambezi River and naturally they had to defend it against indigenous political powers as well as their fellow landowners. The Portuguese state wasn't interested in deploying extensive forces to secure these holdings (and what forces they did provide (100 - 300 men) were poorly trained, motivated and armed) so the landowners turned to military slaves to meet their security needs.

The slaves were settled in strategically located hamlets and kept separate from the rest of the slave population. They were used as soldiers, tax collectors, policemen, ivory hunters and slave raiders. These military slaves, separated from their original societies and kept isolated from larger population, developed their own identities as Chikunda ('conquerors') with their own customs and practices.

Source: Allen Isaacman and Derek Peterson, Making the Chikunda: Military Slavery and Ethnicity in Southern Africa, 1750 - 1900, The International Journal of African Historical Studies Vol 36. No. 2

The Asante also used slave warriors. Jaja of Opobo became King of Opobo. He too was a slave. Slave warriors or slaves who rose to power were prevalent outside the Muslim world to my knowledge in 18th century Bonny and Kalabari(In Modern day Nigeria).

Experts, would Arminius fall into this category?

My understanding is that he was essentially a noble hostage used to maintain German loyalty to Rome's cause. Would it be accurate to describe him in effect a slave that obtained military prestige before turning his back on Rome?