The Varangian Guard originated as settlers and emigres from Northern Europe who traveled southwards and sold their services as mercenaries to the wealthy nobles around the Volga River, the Caspian Sea, and the Black Sea.

Throughout the history of the later Byzantine Empire the emperor became more and more dependent on these recruits from various feudal states in Northern Europe. One example given by M. R. Dawkins (1947):

In 1195 Alexios III, Angelos, alarmed by the threats of the emperor Henry IV, sent three Varangians with golden bulls to ask help of each of the three Scandinavian kings.

Although the Varangian Guard levied troops from all over Europe, from many different religious traditions, the mercenary-bodyguards swore their service to the emperor. The Byzantine Emperor justified himself as the defender of Christ by claiming that he was the successor the first Christian emperor of Constantinople, Constantine.

So since the role of the Emperor was primarily that of defending Constantinople from rulers who were not the successors to Constantine, or who did not have the defense of the Eastern Orthodox church as their priority, then his personal army would follow this same Christian constitution.

The fact that these recruits spoke different languages doesn't necessarily mean that they brought their local traditions to Constantinople, the religious center of the Byzantine Empire. Ecclesiastical figures in Constantinople even translated the Greek liturgy into Danish, Slavonic, English, and Persian so that the Varangians could participate in Byzantine ceremonies (see Dawkins- Later History of the Varangian Guard).

One classic example of a Northern European/viking noble selling his service to the Byzantines is Harald Hardrada, who reigned as King of Norway after amassing enough wealth via plunder while commander of the Varangian Guard. Although he was a secular prince who claimed the thrones of Denmark, England, and later Norway, Harald spent ten years as a Byzantine military commander in Sicily and the Holy Land. There, he led the Byzantine assault against the Arabs and helped protect/gain concessions for Christian pilgrims heading towards Jerusalem.

Despite championing the Byzantine Crusader cause, it is argued by byzantinist Halvor Tjønn that once Harald arrived in Kievan Rus after serving the Varangian Guard, he sold military secrets to the Rus regarding the defenses of Constantinople that directly led to the Rus' attack on Constantinople in 1042.

Here you can sort of see how the religion of the Varangian Guard combined two phenomena from the High Middle Ages--the religious military order and the soldier-of-fortune. Scandinavian warriors were being handpicked to serve in the court of the successor of Constantine and fight to prevent the non-Christian domination of the Holy Land, but at the same time they retained their language, could not inherit land in the Byzantine Empire, and often sold their services to other nobles along the Black Sea/Volga River trade network.

So to summarize the Varangian Guard (the badass elite Russian-Viking bodyguards of the Byzantine Emperor) adopted the beliefs and liturgical practices of Constantinople because that was where they were employed.