No, especially as they were separated in time by about 500 years (Swiss guard was formed in 1506, the Varangian guard (or at least the tradition of Rus mercenaries) in 874.

The idea of having a personal bodyguard of foreigners isn't unique to the Byzantines and Papacy, many rulers had them - leaving aside prestige, foreigners are considered less likely to have divided loyalties with local nobles and the language barrier protects against subversion.

The Swiss were by far the most popular mercenary troops in Europe from the 15th century onwards, providing infantry, and occasionally artillery, usually of Guards status, to France, Spain, Saxony, Bavaria, Naples/Two Sicilies, and, of course, the Papacy, right up until the 19th century. They were well regarded because, as long as they were paid, they remained faithful to their employers, even when national troops didn't, as in Paris in 1792 and 1830. The French employed them extensively from about 1450 onwards, and introduced them to Italy with the French invasion of 1494, they then became popular with the various Italian states, but especially with the papacy.