In short answer, it was a kind of more Danico (in Danish/ Scandinavian ways) to have multiple spouses in the 11th century regardless of their practiced religion, and any king's sons from such kind of multiple relationship were generally eligible as a possible throne candidate at least well into the late 12th century in Scandinavia, well after at least a century after the acceptance of Christianity. In this sense, the hierarchical distinction between the formal wife and concubine from elite family was sometimes not so easy to discern, or scholars have also often difficulty in identifying the exact nature of relationship in individual cases. The "marriage" between Cnut and Ælfgifu of Northampton was certainly one of such difficult cases.
+++
As OP (probably) knows, Bolton (author of one of the latest academic biography of Cnut the great) proposes in the 21th century that their "marriage" could be valid in legal terms as that between Cnut and Emma (Bolton 2007: 53-58) (it had generally regarded as a concubinage contra legal marriage between Cnut and Emma), and this hypothesis has apparently been accepted widely in academic circles (Bolton 2017: 67-71; Lavelle 2017: 10f.). In fact, Bolton proposes two possibility of interpreting their relationship, either:
While Lavelle emphasizes the former, fully legal aspect of their unison (Lavelle 2017: 11), I'm personally leaning into the latter, since a few, especially Scandinavian example of ruler's multiple relationships are known in the 11th century, and it is difficult to judge Ælfgifu's case based on the extant evidence (The most detailed contemporary primary text, Encomium Emmae Reginae ("The praise for Queen Emma"), is clearly vastly pro-Emma, against Ælfgifu source, and the inheritance claim of Ælfgifu's children (Swein in Norway and Harold in Denmark) after Cnut's death would be almost the same between the former and the latter as long as their paternity is acknowledged by the father, Cnut).
The complexity of allegedly multiple relationships of Scandinavian rulers and the royal children born in such relationships between the male ruler and woman of often high social rankings have attracted much attention from women's history as well as historian specialized in political history for more than three decades (Jochens 1987; Nors 1996). In sum, their researches suppose that the concubinage of ruler was not solely motivated the male ruler's arbitrary lust, but also by the need of the political alliances between the ruler and the local elite that the woman came from (Jón Viðar Sigurðsson 2017: 21f.).
In the 11th century, King Harald Hardråde of Norway (d. 1066) and his rival, King Svend Estridsen (d. 1076) are also both known to have practiced such a "concubinage".
A marriage status of King Harald Hardråde is sometimes confusing to modern people (including the game player of Crusader King series) - In later tradition like Heimskringla, while his wife, Elisiv of Kiev, apparently kept married with Harald until his death and even outlived him, both of two sons of Haralds, namely Magnus and Olav, were born between Harald and Tora Torbergsdatter of the Giske family, the local elite of central Norway (Cf. Bandlien 2011: 19), and their succession apparently proceeded with no problem. It is likely that Harald's relationship with Tora of the Giske is also socially important and accepted widely as that between Harald and Elisiv.
Another example, Svend, has some more evidence. The scholion (medieval annotation - in that case, just added a few year after the main text) of Adam of Bremen who met once Svend in person makes a note on his multiple women:
"The most illustrious king of the Danes [Svend Estridsen] was afflicted with incontinence only in respect to women, yet not of his own will, in my opinion, but from racial vice. Nevertheless, he did not escape the vengeance for his evil doing, for one of his concubines, Thore, put his legitimate queen, Gythe, to death with poison. And when King Sven sent Thore's son, named Magnus, to Rome to be consecrated there for the kingship, the unhappy youth, died on the way. After him the wicked mother did not have another son (Scholion 72 (73). The English translation is taken from: [Tschan trans. 2002 (1959): 132).
It's true that the author of the scholion (not necessarily Adam himself) distinguished "legitimate queen" Gythe and "concubine" Thore, but the former does not appear in any other tradition, and even in this scholion, the son between King Svend and Thore, the concubine, was de facto regarded as legitimate/ eligible for the consecration also by Rome. Then, based on the child's status, there was little difference between "legitimate" and "concubinage" relationship again here.
You might also feel that either Adam or the scholion author is too lenient to this apparently "non-Christian" multiple relationship (than generally assumed). The Pope himself was indeed no exception. Svend and (some of many of) his sons were protege of Pope Gregory VII (r. 1073-85), as I illustrated before in: To what degree World the early church of Scandinavia be connected to the church of the continent and the papacy?, and Gregory also alludes to Svend's alleged womanizing practice in the letter addressed to Svend's son, King Harald hen (r. 1076?-1080):
"Even though he [Svend], had let the restriction of his body loosen following the allurement of corruption by the devil's instigation, we don't doubt that he occupies the seat (one of the seats) in the heaven reserved by the supreme God for the favorite rulers (Reg. Greg. V-10 (linked to the original text, and I translate into clumsy English))."
Further research and comparison with the practices across medieval Europe (Cf. Bartlett 2020: 155-86; Rüdiger 2020), especially on the transformation and establishment of the social norm on the marriage and legitimacy, will illustrate the formation of "(pan-)European" aristocratic culture under the possible influence of Christianity in long-term, and this topic will also be interesting in diverse field of history as well, such as political, social, religious as well as family and gender history.
References:
+++
(Edited): fixes typo and format a bit.
Ne na ma wife þonne an hæbbe 7 þæt beo hs beweddode wif, 7 beo be þære anrem þa hwile þe heo libbe, þæs scyldon æfre ge, se þe wyle Godes lage giman mid rihte 7 wiþ hellebyrne beorghan his sawle
Never may a man hold more than one woman as his wedded wife, and there she is his only, while she may live, thus they ever shall, which he wishes to God's law with righteousness, or burn his soul with hellfire.
G Variant of Canute's Winchester Code, promulgated between 1020 and 1021 (Hs. Cotton Nero, A 1)^Translation ^is ^my ^own
There it is, written in simple(ish) Old English what would happen to any man who marries more than one woman! Let's pack it in folks, we got him.
...
Wait, you need more than that?
Well then....
In the Medieval world, like in the world of the Pirates of the Caribbean, the written laws of the people were more like what you'd call guidelines than actual rules. The laws that were written down in the leges law texts of the early Medieval Kingdoms, were not like law codes that we have today. Today, our law codes are legally binding on individuals and can be interpreted and applied by, ostensibly, independent figures who have studied the laws and their application in a rigorous course of study. That was not necessarily the case in the early Medieval world however. The Anglo-Saxon law codes were not intended to be used by judges to consult relevant case law and used to determine the outcome of trials. The proceedings of Anglo-Saxon courts were not like ours, and they did not rely on this overtly legalistic and professionalized administrative system.
The actual application of law was not a neutral and clear system that used legal texts. It was much more fluid, where the legal texts might be used in accordance with local practices, the advice of local councils and notables, the whatever the bishop was pushing for, the influence of the wronged parties, the influence of the ostensible criminal, and so much more. The actual mechanisms of law were fuzzy and not clearly delineated, so the result was kind of mishmash of weird half baked legal systems that were somewhere between the earlier feud based codes and later Roman developments of law systems that emphasized the role of written legal texts that traced legitimacy back to Rome and to the Church.
Now that's kind of a general summary of how law worked, or didn't, in Anglo-Saxon England, so what Canute was doing was clearly, well if illegal is not the right word perhaps something more like, improper. So what gives?
This gets to a tension that was never really adequately resolved in England and in many other parts of Europe following their conversion to Christianity. Prior to Christianization it was very common for high ranking men in societies such as Scandinavia, pre-Christian England, even Christian Ireland, to have multiple women attached to them in a weird status. This often gets described as concubinage, but that doesn't quite do the system, or situation, justice.
And the idea that a king would actually be beholden to the written laws of the land is frankly somewhat laughable. Canute the Great was not even the only English monarch of the 11th century to have more than one wife! Nor was monogamy, despite the teachings of the Church and the clear influence of Christian/Roman preferences for monogamy, ubiquitous even in areas of Europe that had been Christianized for longer than England and Scandinavia!
/u/y_sengaku deals with the other Scandinavian monarchs who had multiple wives/concubines/women of unclear status, so I will deal with the specifically English situations. Canute was not the only King of England to hold multiple women in some form of marriage, several decades after Canute's death, Harold Godwinson likewise held multiple women in marriage, similar to other Scandinavian rulers.
So was this the result of a Scandinavian cultural practice bleeding into the practices of the upper echelons of Anglo-Saxon society? Perhaps, but that does not necessarily explain the relative abundance of women living in quasi married status in the Frankish court a little earlier in time. Instead, I think that we need to rethink a lot of these categories, marriage, husband, wife, and so on in more nebulous degrees than our modern sensibilities would like, and accept that the "laws" of the time period were not indicative of royal practices and the needs of political expediency, as well as the whims of the particular monarch in question.
There is reason to suspect though that the "unmarriages", as termed by Ruth Mazo Karras, of figures like Canute, Harold, and others, were partly a result of holdover from pre-Christian times as well as fuzzily understood but also stemming from broadly accepted living situations that were common throughout the Middle Ages. Indeed much of Medieval marriage in general is very hazy according to Karras and our desire to categorize and describe the various living, and sleeping, arrangements of medieval figures is going to be frustrated as there never really were simple and clear delineations on what constituted marriage and what did not. For example, the mother of Charlemagne's son Pippin the Hunchback was a woman is often called a concubine, but was not enslaved, indeed she was a high status woman whom Charlemagne did eventually marry in a more formalized ceremony, but their relationship was known to the court long before that. Charlemagne's daughters likewise engaged in some of this unofficial but sanctioned relationship building with their social peers.
Now Canute is unique, compared to say Harold, in that he seems to have been wedded twice, but what did that status actually entail? Was it permanent? Could it be rescinded? These are good questions that we don't necessarily have wholly satisfactory answers to, and I do think that /u/y_sengaku's connection to the increased oversight of the Church over the Middle Ages, and the connections to Scandinavian practices, hold true, but it is worth noting that this was not uniquel Scandinavian. Kings holding multiple women as quasi-wives was not an unusual phenomenon for the early Middle Ages, and that despite the presence of laws to the contrary, the behavior of kings in this regard, marriage and monogamy, at this time was more or less unconstrained by Church influence.
Why was this the case?
The Papacy at this point in history, was not the institution that we know today, or even the Papacy of the later Middle Ages. The situation in the 11th century, with loose control by Rome but in theory subservience to the Pope, was not conducive to the major ecclesiastical figures actually being able to enforce Church doctrine and teachings on the high nobility of Northern Europe. The ability of the Papacy to actually exert control over the Christian communities of Europe in the early Middle Ages, lets say up until the first crusade for a benchmark, was constantly called into question. The Popes indeed were often at the mercy of many political actors of the early Middle Ages, and alliances that the Papal states made with powers such as the Carolingians should be seen as evidence of the intense weakness of the Papacy to actually conduct itself as a power broker in much of Western Europe, to say nothing of the eastern Mediterranean where the interest of the Byzantine emperors in the affairs of the Latin Church was, to understate it, thin.
All of this means that the Church in Rome was not able to effectively enforce orthodox (not Orthodox) practices on the wide distribution of Christendom at the time. This took many forms, and the way that is relevant for us relates to monogamous marriage. The Church just simply did not have the power to actually hold the monarchs, like Canute or Charlemagne, to task on these kinds of matters, regardless of the levels of piety by the monarchs or by the people at large.