As usual, the first thing that needs to be pointed out is that the Middle Ages is a period that lasted for 1000 years and Europe is a rather large area that happened to see quite significant population shifts between 5th and 15th century. I'm not deeply familiar with the game, but it seems it is not set in any particular time period, freely mixing the artifacts from somewhere between late 9th and early 16th century and thus we cannot limit ourselves to any more specific timeframe. That said, let's move the matter at hand. The short answer seems to be very simple, as the medieval Europe was predominantly white, but the devil, as always, is in the details.

First and foremost, the definition of 'white' is not as uniform as one may think, mainly due to various receptions of the relatively new notion of the 'human race' that have arisen in mid-18th century Europe and then developed quite differently in various parts of the world. What is important to note is that the most common division proposed by Blumenbach in 1775 included five main 'races': European, African, Asian, Indian (American) and Malaysian (Austronesian). This was sometimes reduced to three main 'races' with Austronesians and American Indians being considered a subtypes of the 'Asian race'. What is important to note here is that the skin colour was not a most important or sometimes even essential part of the classification in comparison with the other phylogenetic traits, what is most blatantly seen in the case of Indians who, despite having distinctly darker skin, were classified as 'Europeid' (i.e. white). This classification, according to which anyone who does not show any obvious African or East Asian traits is simply considered 'white' seems to be prevalent in Europe, at least in part thanks to the high ethnic millennia-old diversity in the area. The construct of the 'race' in Unites States however can vary, given the prevalence of the people having Anglo-Saxon and Celtic origins in the initial phases of the country development that, compounded with strict racial policies, caused the term 'white' to be associated primarily with people inhabiting Central and Northern Europe (Anglo-Saxons, Germans, Celts, Slavs, Scandinavians and Balts) with other Europeans being considered 'racially' different from them.

Now, when we are speaking of 'darker skin colours', assuming that we are meaning people who do not show visible phylogenetic traits typical to African and East Asian peoples (i.e. 'white people' from the standpoint of both modern Europeans and 19th century ethnologists), then the diversity brought by the existence of various ethnicities should only be expected, as beside people that are universally considered 'white', such as the groups I mentioned earlier, we need to include Italians, Greeks, Jews, Arabs, Armenians, Persians, Turks and Berbers at the very least. This would be quite visible in Spain that, until the late 15th century, has been a Muslim country, inhabited by descendants of Celtiberians, Phoenicians, Romans, Arabs and Berbers. Somewhere around 11th century a new distinct Dravidic Indian minority came to Europe where it came to be known as 'Gypsies'. It should also be pointed that the Europe was more of a patchwork rather than a melting pot, as in the mid-to-high Middle Ages various cultural groups were not readily mixing for cultural and religious reasons and partially due to lack of mobility, at least in comparison with what we can observe today.

Now, what about people who could not have been considered 'white' by modern standards? The people of African descent, although most definitely not completely absent, were exceedingly rare almost to a point that even a very experienced person would have an extremely slim chance to meet one in Europe. This chance, however, became somewhat higher for the inhabitants of the Muslim lands, especially the Middle East due to the proximity to the Nubia and Ethiopia. In late Middle Ages this would also apply to the Northwestern Africa that had contacts with the Mali Empire. On the other hand, the Ethiopians maintained relationships with the Eastern Roman Empire, as evidenced by the accounts of Robert de Clari, a Picardian knight who participated in the Fourth Crusade and who mentions that Emperor Manuel I was visited by an 'Ethiopian king' in mid-1170s.

The most obvious non-white people in Medieval Europe would be, inexplicably rarely mentioned but in fact very numerous Central and East Asians. The most obvious groups would have been various Altai peoples of Polovtsian-Kipchak Confederation and, of course, the Mongols. The former arrived somewhere in 11th century, conquering the entire area that is now southern Russia and Ukraine, overpowering southern areas of Kievan Rus and raiding eastern areas of Poland and Hungary until they have been stopped by the Hungarian king Laszlo I and later pushed behind Don by Vsevolod I. They have been finally defeated by Mongols who arrived in Europe in 13th century, subjugated the Kievan Rus and managed to move as far west as Western Poland, close to the German border. For many centuries, Mongols have been holding the areas encompassing modern southern Ukraine, European Russia, Moldova and Eastern Romania while controlling the remaining parts of the Kievan Rus. Their relations with other countries were also not entirely hostile, as there are accounts of Mongols allying themselves with other powers in the area, such as Tokhtamysh who allied himself with Lithuanian ruler Vytautas in an attempt to regain the leadership of the Golden Horde (after the defeat at Vorskla river in 1399 those loyal to him withdrew to Lithuania where they were allowed to settle) or Tokhtamysh's son, Jalal ad-Din whose unit fought together with Poles and Lithuanians against Teutonic Order and its allies in the Battle of Tannenberg.

To sum this up, the medieval Europe was a rather diverse place, and although the specific ethic groups were usually large and definitely not uniformly distributed across the continent, they have existed nevertheless, with Central Asians being easily the largest 'non-white' group while Arabs and Turks were the largest ethnic groups hailing from outside Europe.

EDIT: Fixed a few typographic errors.