Edit: this post was apparently too long to post as one, so I broke it in two and posted the second part as a reply to the first. I probably could edit it for size, but since I already spent the time writing it, hopefully I'll be forgiven for not revising it so extensively as to cut it in half.

Archaeology PhD here with a regional focus on Ethiopian archaeology. I'll try to weigh in on your question, though it's not a question I generally regard directly in my research, so forgive me for treading lightly on some aspects of your question and not having a full list of citations at hand.

TL;DR: Ethiopia has long been at the center or key periphery to trade and geopolitics in the world, particularly during the spread of Christianity and Islam out of the Eastern Mediterranean and Middle East. Like neighboring Sudan, all three major Abrahamic faiths have had some presence there. However, the phrasing of your question makes it sound like you are asking why do they all get along? In fact, though for larger stretches of Ethiopian history, the three religions have not been personally antagonistic to one another, economic and political interests often used religion as a cover to persecute one another.

First off, you have to realize that Ethiopia was effectively at an important cross-road of civilizations during Classical Antiquity and the early Medieval Period, particularly in geographical relation to the Abrahamic faiths. Ethiopia was a major player in Red Sea trade before and especially during the Aksumite Period (c. 0 - 700 AD). In archaeology, we have found everything from Roman wine amphora in Aksumite tombs to Aksumite coins in India. The Periplus of the Erythrean Sea, a first century AD itinerary of coastal trade communities around the Red Sea Indian Ocean even mentioned Aksum's port city of Adulis and the goods it imported and exported. With the spread of Islam, global trade networks shift and much of the Horn comes under the control of Muslim states, but Ethiopia remained an important source of materials nonetheless. For what it's worth, many traditional Ethiopian populations (i.e. Ethiopians off the ancient state) also spoke Semitic languages (e.g. Amharic and Tigrinya), related to Aramaic, Hebrew, and Arabic.

To your question more directly, I would be careful how you phrase it. The Judaism, Christianity, and Islam where often at odds in Ethiopian history, though as often as not this antagonism was about economics and political power as much if not more than it was about religion.

Christianity formally arrives in Ethiopia in the 4th century when two young Christians supposedly wreck on the shores of modern Eritrea and are taken as captives/slaves to the Aksumite court. The youths become companions to the royal heir Ezana, then in his youth. The three grow up together, Ezana converts to Christianity following the proselytizing of his companions, and he asks one of them, Frumentius, to go to Alexandria, head of the Coptic Church, to ask for a bishop so that priests can be trained and appointed in Ethiopia and other Christians and converts can be ministered to. Frumentius is lauded by the Alexandrian patriarch for his work and is made the first Metropolitan (bishop, basically) of Aksum. We know Ezana actually converted by two means. First, though I've never really pursued the original documents, is that the Alexandrian patriarchy supposedly has independent records of this appointment. More tangibly, though, Ezana, like many Aksumite kings, minted his own coins to the same standard as the (Eastern) Roman Empire. His earlier coins contain pagan iconography, while his later coins feature a cross and a Ge'ez inscription that basically translates as "by this sign you will conquer" in a clear imitation of Constantine's conversion story. A lot of evidence indicates this conversion was probably socially top-down and for economic purposes unlike the Roman Empire. Ezana's coinage, again, was meant to be fungible with Roman coinage to facilitate international trade. Likewise, a stele erected by Ezana found at Aksum contains a Greek inscription and a Ge'ez inscription. Both are largely the same, except for some clever word use that linguists (Tamrat 1972 - Church and State in Ethiopia is always a good source for this basic scholarship) have pointed out: The Ge'ez makes use of a rather banal and generic address for the God of Ezana that was used rather imprecisely by his pagan predecessors for the patron deity of the king; the Greek translation, on the other hand, uses explicitly Christian language that any Eastern Mediterranean Christian would have noticed. This suggests that while there may have been Christians in Ethiopia prior to Ezana's conversion, the conversion was largely an economic move to identify more closely with the Mediterranean world, Ethiopia's prime market region. Cynicism aside, Christianity flourishes.

The by 14th century, a manuscript circulates called the "Kebre Negast" or "Glory of the Kings." The manuscript describes itself as a Coptic document translated from Arabic into Ge'ez, but I think most scholars believe it is really a conflation of disparate writings, myths, and oral histories compiled together. The story it tells is rather interesting, but to keep it brief, it basically says that an early Aksumite queen heard of the glories of King Solomon, visited him, had a child by him, and the Covenant of God was passed on to that son, who returned to Solomon's court and eventually returned to Ethiopia with the symbol of God's covenant, the Ark. The story thus establishes a biblical tie between Ethiopia and Jews/Christians and, not coincidentally, provides a foundation story legitimizing the newly emergent "Solomonic" Dynasty that came to power by overthrowing the previous dynasty claiming that they, not the Zagwe kings, where the true inheritors of the throne of Aksum and thus the divine bloodline traced back to King Solomon.

Islam in Ethiopia has an interesting history as well. According to both Muslim hadith and Ethiopian tradition, the first followers of Muhammad faced persecution in Mecca and were forced to flee. The Ethiopian king at the time supposedly learned of the persecution of people believing in the Abrahamic god across the Red Sea and offered Muhammad's followers sanctuary in Ethiopia until Muhammad was finally able to subdue his persecutors. By the 8th century, Islam spread rapidly across North and East Africa. This spread was not always by adoption and Ethiopia eventually lost control of its coastal territories in modern Eritrea to independent Islamic states. Ethiopia thus went from a primary trading state to a peripheral state that engaged in international trade largely through intermediation by its Muslim neighbors. A number of Islamic states and cities ringed Ethiopia to the north, east and southwest and Muslim caravans became a vital means of international trade for the king and state.

By the 15th century, however, relations between Ethiopia and the states to its southeast like Ifat and Adal, became heated. The Christian and Muslim states regularly raided one another, and some Ethiopian kings demanded their citizens make visible shows of faith by bearing crosses at all times, even going so far as to have them tattooed on their heads or hands (a tradition taht persists today). I want to say it was Zara Yeqob or Amda Seyon ordered this according to their chronicle, but I can't remember. Though often expressed under the guise of religious animosity, it's pretty clear the goals of much of this conflict were at their heart economic. Ethiopia wanted direct access to the sea again (which they would eventually do by reclaiming Eritrea as a semi-autonomous state) while neighboring Muslim states did not want to go through Ethiopia as an intermediary to access its resources of gold, ivory, animal skins, etc... Conflict between Ifat/Adal and Ethiopia is a perennial problem in royal chronicles until the rise of Ahmad ibn Ibrahim al-Ghazi ("Gran": The Left Handed). Ahmad was basically a rabble rouser who managed to become an imam and usurp the power of the Adal sultinate, claiming it for himself and conquering his neighbors. Under the guise of a jihad he then goes on a campaign of conquest in Ethiopia, very nearly succeeding. The contemporary account, the Futuh Al-Habash records his exploits and the focus on looting treasure is notable. Gran nearly succeeds in conquering Ethiopia, offing a number of kings and very nearly taking the spiritual capital of Aksum when well-armed Portuguese explorers and Jesuit missionaries just happen to arrive and save the day. Gran is killed, Adal largely slides into irrelevance to Ethiopia, and Ethiopian attention turns toward the newly arrived Catholic missionaries. Relations between Christians and Muslims in Ethiopia at this point largely return to normalcy with little recorded animosity or inter-state conflict after this, again, probably because the issue was always one more of economic interests than religious.