I asked this question entirely forgetting about the rest of the world, since my main focus was on Western Europe. But hey, even the reasons behind that were answered!
The Roman Catholic church did not have anything close to a pre-Protestant monopoly on Christianity. The major Christian churches were:
The Roman Catholic Church.
The Orthodox Church AKA Eastern Orthodox AKA Orthodox Catholic Church, headed by the Patriarch at Byzantium, and after 1453, this was claimed by the Patriarch of Moscow.
The Church of the East AKA Nestorian Church AKA Assyrian Church, centred on Persia until the late Middle Ages when their centre of gravity moved to Iraq and Syria. Christology: Dyophysite (Nestorian).
The Oriental Orthodox Church, headed by the Patriarch of Alexandria AKA the Coptic Orthodox Pope of Alexandria. Christology: Miaphysite. Primarily a church of Egypt, Ethiopia/Eritrea, and Armenia.
The Nestorian and Oriental Orthodox churches split from the Catholic-Orthodox churches on theological grounds, with the big issue being the nature of Christ. The Nestorians split following the Council of Ephesus in 431, and the Oriental Orthodox Church split following the Council of Chalcedon in 451 when the Pope of Alexandria and the other 13 Egyptian bishops thought that the Chalcedonian Definition was too Nestorian.
The Roman Catholic and the Eastern Orthodox (Orthodox Catholic) churches are both Chalcedonian, and split relatively late (officially in 1054), largely over political rather than theological issues: the authority of the Pope (i.e., the Bishop of Rome).
Each of these churches was the dominant Christian church in its own region: Roman Catholic in Western Europe, Eastern Orthodox in Eastern Europe, Oriental Orthodox in Egypt and Ethiopia, and Nestorian over most of Asia, from Mesopotamia through to China. The Nestorian Church was the largest in geographical extent, but Christians were a minority in most of that area, and they counted perhaps only 1/6 of all Christians within their church.
The dominance of the Roman Catholic and Orthodox churches in their regions owes a lot to state support. In the case of the Orthodox Church, this was an alliance between the Byzantine state and the church. The state benefited from legitimisation and social support by the church, and the church could benefit from state muscle being used suppress rival religions. Both church and state could provide political and financial support for each other. The Roman Catholic Church was part of this arrangement until the mid-8th century, when Byzantium lost control of Ravenna and Rome. Byzantium was promptly replaced by Francia as the controller of Rome, and what had been a Catholic-Orthodox-Byzantine alliance now continued as separate Catholic-Frankish and Orthodox-Byzantine alliances, on the path to the Great Schism of 1054. The importance of the state in supporting the state religion, and keeping it the dominant religion, is shown in the Protestant Reformation, e.g., by the Peace of Augsburg of 1555 which allowed rulers to make their states either Catholic or Lutheran - "cuius regio, eius religio".
The short answer is that it never had the “monopoly.” The Orthodox Church was the dominant Christian religion in the east and the Coptic churches were prevalent in parts of Africa. The Roman Catholic Church had near total dominance in Europe because of shrewd political maneuvering by the popes.
In late antiquity, all the bishops in Europe and the eastern Mediterranean were technically equal, but since Jesus gave Peter the keys to the Church and referred to him as “the rock” of the Church, the Bishops of Rome thought of themselves as “first among equals,” and there is some evidence that the other bishops agreed with that consensus.
After the fall of the western Roman Empire, Christianity was also divided between the east where the patriarch of Constantinople became very powerful and the west where the Roman Church was able to consolidate a lot of power in the vacuum left by the dominate.
In the early Middle Ages, a watershed moment came in 751 when Pope Steven II turned to the Franks for help protecting Rome from invaders instead of the Byzantines, who were perceived to have been weakened by Islamic conquests.
Then in 800 AD, Pope Gregory crowned Charlemagne Roman Emperor, which was huge because if you had asked the Byzantine Emperor what his title was, he would have told you “Roman Emperor.” Gregory established the precedent that only the Bishop of Rome could crown the Roman Emperor. Gregory then worked with Charlemagne to disseminate the Latin rite throughout the Carolingian empire, which at the time was most of continental Europe, and would continue to grow. This is the moment that Roman Catholicism became the mainstay of Europe. Before this point the people were Christian, but it was a crap shoot as to whether they were practicing the Latin rite, Byzantine rite, or some other local form.
The definitive split between east and west came in 1054. The previous year, Pope Leo IX, told the Greek churches in Southern Italy to either adopt the Latin rite or close. In retaliation, the patriarch of Constantinople closed all the Latin churches in Constantinople. Leo then sent a legate to Constantinople to demand that the patriarch recognize Papal supremacy, and the patriarch refused, so the legate excommunicated him, and then the patriarch excommunicated the legate.
Although the Roman pope wasn’t excommunicated, the damage was done, and after this point the two churches split along cultural, liturgical, and theological lines. They also consolidated their grip on the dioceses in their respective rites and the rest is history.
So it wasn’t a particular aspect of Catholicism that made it the main religion of Europe, but power-hungry popes who took every opportunity to exert authority over the larger church. At the expense of the power of the other bishops.