[Part 1]:
Tl; dr: It had been a kind of dowry from Norwegian dynasty to the Kalmar Union, just like Iceland.
1: From the Norse settlement (in the late 9th century?) to the 12th century
"There is a man called Grímr kamban (the Lame). He was the first person to settle the Faroes. For in the time of Haraldr hinn hárfagri (the Fine-Haired), a large number of people fled in the face of his tyranny. Some settled in the Faroes and dwelt there, while some sought other unpopulated lands. Auðr hin djúpauðga sailed out to Iceland and called in at the Faroes and there gave Þorsteinn Red’s daughter Ólof in marriage, and thence originated the greatest family line of the Faroe Islanders, which they call Gata-Chaps, who lived on Austrey (Færeyinga saga, Chap. 1. The translation is taken from: [Faulkes trans. 2016: 5)."
The saga of the Faroe islanders (Færeyinga saga), now only extant in a few compilation of Norwegian royal biographies (kings' sagas) but apparently originated from the early 13th century, begins with these passages. It is a kind of the Faroe counterpart of the origin myth of the Norse-Icelandic settlers. Just as the Icelanders, the author attempts to justify the historical background of the ancestors of the Faroe Islanders as refugees from the tyranny in Norway under the reign of Harald Fairhair (Haraldr hinn hárfagri), legendary monarch of Norway who allegedly unified Norway and became the founding father of the medieval dynasty. As I explained before in: Did Harald Fine/Fairhair actually exist?, 'real' historical model of Harald Finehair, Haraldr lúfa, a ruler of SW Norway around 900, probably had little to nothing to do with the initial settlers, and the political influence of him and his sons was probably also limited to the coastal region of western and/or SW Norway. Auðr 'the Deep-Minded' was also a kind of legendary female Icelandic settler prominently featured in the famous saga of Icelanders, Laxdœla saga.
The most famous of these biographical compilation, Heimskringla, also narrates the Christianization as well as the formal subjugation (by the payment of tribute) of of the Faroe chieftains largely under the influence of another quasi-legendary Norwegian monarch, St. Olaf (d. 1030) (the saga of St. Olaf, mainly Chaps. 124-129, 135f., 143. English translation is found in [Finlay & Faulkes trans. 2014: 142-47, 157-60, 176-79]). The saga of the Faroe islanders tell us a bit different story - It was not St. Olaf (sometimes called as Olaf II), but another, preceding namesake Norwegian monarch, Olaf Tryggvason (d. 1000) who played a crucial role (also just as the case of Iceland in later Icelander's story) (Færeyinga saga, Chaps. 28-35. Cf. [Faulkes trans. 2016: 37-44]). In later, medieval Icelandic historical writings, this another (first?) Olaf had got popular as the key figure in the Christianization of Iceland, as I also recently illustrated in: Everybody loves Olaf Tryggvason. Why?. While few historians doubts the historicity of these two Olafs, however, the majority of them also agree now that their alleged deed in the sagas had been mostly an embellishment, and we don't really know how the 'real' story was like. Anyway, the saga (the saga of St. Olaf in Heimskringla) also admits that St. Olaf finally failed in imposing the tax to the Faroe islanders in the end.
These conflicting narratives of early history of 'Norse' Faroe Islands (I deliberately omit the issue here whether there had been any settlers before the the settlement of the Norse people) show saga authors try to presents the Faroe Islands as a kind of another Iceland, a part of larger '(Old) Norse World' across the North Atlantic that shared the basic historical-cultural element (Norse settlers during the Viking Age, Old Norse as spoken language), and with a very loose tie with their alleged original homeland, Norway. They should be seen primarily as a retrospective historical narrative from the 13th century when the crown of Norway was extending their political hegemony in the North Atlantic Isles, thus transforming the cultural 'Old Norse World' into their own 'Norwegian Realm (Dominion)'.
2: 'Norse' Faroe Islands depicted in the contemporary text in the 12th Century
Roughly speaking, (late 11th and) early 12th century was the re-organization and establishment phase of the church hierarchy (diocesan organization) in post Viking-Age Scandinavia, and Danish and German metropolitan authority contested each other also for the affiliation of North Atlantic as well as Scandinavian suffragans, including the bishop of Faroe Islands.
In the summer of 1139, Bishop Orm(r) of the Faroe Islands was listed among the participant of the provincial church council presided by Archbishop Eskil of Lund (now Sweden, but Denmark in the Middle Ages), with the attendance of visiting papal legate. This is the almost oldest contemporary written evidence on the connection between Scandinavian mainland and the Faroe Islands, but we don't now what this bishop Orm(r) really did in his alleged diocese on site. With the separation of Norwegian church province of Trondheim(-Nidaros) from that of Lund in the beginning of the 1150s, the suffragan bishops in the North Atlantic Isles, such as Icelandic and Faroe ones, were then transferred under the authority of the Norwegian archbishop of Trondheim (also called Nidaros) - though their real influence seemed to be not so much, just as the loose political connection with the crown of Norway.
History of Norway (Historia Norwegie), customarily dated to the 3rd half of the 12th century and possibly connected with this newly foundation of Norwegian-North Atlantic church province, describes the Faroe Islands as following:
"In the streams of ocean there are also 'islands of sheep', eighteen in number, which inhabitants call Færeyjar in their native tongue, for fat flocks abound in the ownership of the farmers there, some having sheep by the thousand. These islanders also pay tribute to our kings at fixed times (English translation is taken from: [Kunin trans. 2001: 10])."
The Faroe Islands in this text belonged to the category of 'tributary islands' in the West Sea [to Norway], just as the Orkneys and the 'southern' islands (now called Western Isles [to Scotland]), but researchers agree that even at that phase [in the middle 12th century] the payment of islanders to the crown to Norway was not the regular tax, but at most occasional tribute.
3: A lesser clergy from the Faroe islands 'usurped' a Norwegian Crown - the Faroe Islands under the Sverrir (Sverre) Dynasty of Norway (1177/85-1319)
Post-Viking Age Norway suffered from the incessant strife among the throne claimants from 1130 to 1240. An arrival of a clergy from the Faroe islands in 1176, however, turned the tide of this Norwegian medieval 'Civil War' period. He was elected as an leader/anti-king of the factional rebellion against King Magnus Erlingsson (d. 1184), called Birkebeiner ('birch-leg(warmer)s') in 1177, and after several victories, succeeded in unifying Norway (though temporary) as a sole king in the middle of the 1180s. His name is Sverre (Sverrir in Old Norse) (d. 1202), and he claimed himself as a bastard of late King Sigurd Munn (d. 1155 - notes that there were many kings in Norway in the 12th century Norway, sometimes king"s" at the same time), and had been a protege of the bishop of the Faroe Islands in his saga, officially 'supervised' by him. He also later got into conflict further with the Norwegian archbishop(s!) and banned/ got imposed the interdiction by the Pope, just as his contemporary European kings like King John of England and King Philip II of France. After his and his son's death, the throne strife still lingered in the first decades of the 13th century, but his grandson King Håkon [IV] Håkonsson ("the Old") (r. 1217-63) finally put an end with the throne strife in the end, and extended his political influence across the North Atlantic (his subjugation of Iceland and Greenland was famous). Thus, the reign of Sverrir Dynasty (1177/85 as a sole king to 1319) was also, so to speak, a golden period for medieval Norway in the 19th century nationalistic Norwegian historiography.
Modern historians are now generally skeptical of this Sverre's claim of alleged royal blood, however (Cf. Krag 2005). If we accept their re-evaluation of relevant texts, he was probably not in fact a king's bastard, and had not been claimed in public as so when he visited in Norway at first. Pope Innocent III called Sverre as a 'regecide' as well as 'royal pretender' in his letter, and this accusation might have been correct after all. Anyway, the man who came from the Faroe Islands (Sverre had migrated to the islands when he was 5 years old, together with his mother, so technically he had been born in Norway - if we believe his 'official' saga), not legendary Harald Fairhair, became the founding father of the dynasty that ruled the wider 'Norwegian Realm' from Greenland to Scandinavian Peninsula as well as form the Isle of Man to the Arctic Norway, across the North Atlantic.
[Continue to be Part 2]