The military orders were, almost entirely, a uniquely Latin (later Catholic) Christian institution. There were a few instances when Protestant Christianity dabbled with the concept but no other Christian denomination or other religion developed something quite like the medieval military orders (no matter what CKII might say).
The three greatest orders, the Templars, Teutonic Knights, and the Hospitallers, are a useful example of what we can define as a military order. All three were religious fraternities whose brethren were bound by the three traditional monastic vows: chastity, poverty, and obedience. They swore these vows for life. Only expulsion from their order or being granted permission to transfer to a more austere order could release them from this. These three orders also maintained a military role, defending Christian pilgrims from attack and conducting campaigns against non-Christians. Two of them, the Teutonic Knights and the Hospitallers, had a hospitaller role also, providing medical or charitable care. However, this was not an integral trait for military orders and not all orders maintained a hospitaller role (e.g. the Templars).
In terms of membership, these three orders had a tripartite structure of knights, sergeants, and chaplains. Knight-brethren had to be of knightly background, a peasant could not join the Hospitallers and be 'promoted' to knight, for example. These men made up the bulk of the order's leadership. Sergeants were non-knightly and a mixture of military and administrative personnel. Chaplains were the order's priestly contingent. Each order was also heavily centralised, with its different priories and houses across Europe and the Levant all reporting to a single grandmaster alongside a council of officers or a general chapter that could dictate or influence elections, promotions, and financial and military strategy in these priories. This level of central control puts it in great contrast to many of the monastic orders whose houses could often decide these matters for themselves (though this centralisation was not unique to the military orders and many religious orders became more centralised in the late medieval period). Finally, all three orders were subordinate to, and recognised and endorsed by, the Papacy.
So from these three orders, the largest and most famous of the medieval military orders, including the two first orders (the Templars and Hospitallers), we can define the medieval military order as so:
I have yet to come across any other institution outside Catholic Christianity that quite matches these traits. The lack of any centralised religious hierarchy or traditions of religious poverty and chastity in the various medieval European pagan religions precluded the development of military orders. Orthodox Christianity also avoided forming these orders, though the Order of the Dragon, a chivalric order that was influenced by the military orders (such as having holy war as an objective), did have a few Orthodox members. There are Asian traditions of warrior-monks, but these do not fulfill the traits of the Christian military order (as far as I can tell, I am happy to be corrected by someone better versed in Asian religious history). The Ikko Ikki of Japan, for example, were not equivalent to a knightly organisation, were not under central church control, and were not a centralised single institution in the same way as a military order.
There has been some attempt to link the military orders to the Muslim ribat, with some historians going so far as to claim the ribat as the inspiration for the Christian military orders. However, closer study has found that these two institutions were not very similar. A ribat could refer to a range of different institutions, including a hospice or a frontier fortress. It was not necessarily a military institution and was certainly not one on the same scale as a military order. Membership was temporary, again differentiating it from the perpetual vows brethren of the military orders swore. Finally, there is no evidence of a wider organisation coordinating these ribats in the same way that the preceptories and castles of the Hospitallers were all part of a wider order.
Anglican Christianity did try to revive the military order a couple of times. In 1540 Henry VIII's council in Ireland proposed creating a new order to police Leinster. This proposal maintained some of the trappings of the medieval military orders. For example, apart from the first thirteen founder-brethren, all subsequent members would swear vows of celibacy. However, nothing came of this. There were attempts again in the seventeenth century to resurrect the Hospitallers in England and in the nineteenth century a Hospitaller successor order was instituted and then recognised by the crown and is still around today. However, this order, the Venerable Order of St John, has no military role only a charitable one (though the rank of knight is still used for some members).
I should mention that there were some exceptions to this model of the military orders that I have given above. Brethren in the Order of Santiago were vowed only to conjugal chastity, they could marry. The thirteenth-century Order of the Blessed Virgin Mary did not fight holy war but was instead deployed to northern Italy to deal with the conflict between the Guelph and Ghibelline parties. However, generally such orders are the exception, not the rule, at least in the medieval period. The orders of Calatrava, St Thomas of Acre, St Lazarus, and the Sword Brethren all fulfill this model of a 'typical' military order.
Sources:
Malcolm Barber, The New Knighthood (Cambridge, 1994).
Enrique Gallego Blanco (ed.), The Rule of the Spanish Military Order of St. James 1170-1493 (Leiden, 1971)
Alan Forey, ‘The Military Orders and Holy War against Christians in the Thirteenth Century’, The English Historical Review, Vol. 104, No. 410 (Jan., 1989).
Alan Forey, 'The Emergence of the Military Order in the Twelfth Century', Journal of Ecclesiastical History, Vol. 36 (1985).
Nicholas Morton, The Teutonic Knights in the Holy Land (Woodbridge, 2009)
Jonathan Riley-Smith, ‘Towards a History of the Military-Religious Orders’, in Karl Borchardt, Nikolas Jaspert, and Helen J. Nicholson (eds), The Hospitallers, the Mediterranean and Europe: Festschrift for Anthony Luttrell (Aldershot, 2007).
Jonathan Riley-Smith, The Knights Hospitaller in the Levant, c.1070-1309 (Basingstoke, 2012)
State Papers: published under the authority of His Majesty’s Commission. King Henry the Eighth. (London, 1830-52), iii.