The status of Norway in early modern period was certainly was a bit tricky, especially 'seen by other European powers' part in OP's question. On the other hand, we have good amount of contemporary sources that confirm the continuous existence of the kingdom of Norway, still with its own laws and administrative organs (Jespersen in Jespersen ed. 2000: 47), even after the promulgation of the famous 'Norwegian paragraph' in the coronation charter of King Christian III of Denmark in 1536 stating that Norway 'hereafter should and will also be under the Crown of Denmark, just as one of other regions of the kingdoms, such as Jylland, Fyn, Sjælland or Skåne, and hereafter should be not or be not called as its own kingdom, but a part of the kingdom of Denmark and forever under the Crown of Denmark'.

To give an obvious example, the official title of Oldenburg dynasty's king of Denmark from the 16th to the beginning of the 19th century was always as that: '......by the grace of the God, king of Denmark, Norway, the Wends as well as the Goths (Götar)......', as also stated in the prologue of King Christian's IV's Norwegian Law in 1604. 'Heir of Norway' became also customary title for the king's sons of the Oldenburg royal family in 16th and 17th century (Rian in Bjørgo, Rian & Kaartvedt 1995: 141f.). Even the Danish council of the realm, the representative of Danish aristocracy of the time, called their kingdom as 'Denmark and Norway with their dependent lands' during the interregnum period in the eve of the peace of Westphalia (1648) (Jespersen in Jespersen 2000: 47). These testimonies can refute our lingering doubt of Dano-Norwegian union only as an retrospective invention by modern historians enough, I suppose.

Then, why did King Christian III of Denmark (who had also involved himself with the wording of the coronation charter) did not put the 'Norwegian paragraph' into practice in the end that would lead to the constitutional dissolution of the kingdom of Norway as an entity? Modern researchers regard the complicated dynastic succession circumstances in the first half of the 16th century as a key to interpret this kinda mystery.

It is true that King Christian III had just won the throne of Denmark after Count's Feud (Succession War called Grevens fejde in Danish) in 1536 when he codified 'the Norwegian paragraph', but his position within Denmark-Norwegian union was far from secure: While the Norwegian Council of the Realm had stopped to function, the former top of this council, Archbishop Olaf Engelbrektsson was still alive and stayed in Norway, and Christian's archenemy, (ex-)King Christian II of Denmark (d. 1559) was also alive as an exile in the Low countries and got support from his brother-in-law, Emperor Charles V of HRE. he also had two daughters, Countess Dorothea of Pfalz and Duchess Christine of Lotharingia. They were not so willing to give up the claim of succession to all the kingdoms of former Kalmar union (Denmark, Sweden and Norway), just as King Gustaf Vasa of Sweden, in the middle of the 16th century, though Christian II, first among them, agreed to part with his claim over 'the kingdom of Denmark and Norway' in 1546 (Rian in Bjørgo, Rian & Kaartvedt 1995: 138). To cope with such external intervention into Scandinavia on the pleas of heir to Norway, it would have been not so wise to eliminate the title of 'kingdom of Norway' entirely.

References:

• Bjørgo, Narve, Øystein Rian & Alf Kaartvedt, Norsk utenrikspolitikks historie, i: Selvstendighet og union fra middelalderen til 1905. Oslo: Universitetsforlaget, 1995.
• Jespersen, Leon. 'The Constitutional and Administrative Situation'. In: Id. (ed.), A Revolution from Above? The Power State of 16th and 17th Century Scandinavia, pp. 44-65. Odense: Odense UP, 2000.
• Rian, Øystein. 'Norges status som kongerike 1536–1814'. Norgeshistorie.no (in Norwegian: University of Oslo)
• Sandmo, Erling. 'Selvstendighetstap og foreningstid' Norgeshistorie.no (in Norwegian: University of Oslo)

The standard reference was definitely to Denmark and the King of Denmark, which we can see in English sources from Christian IV's visit to England in 1606. However one reference to Anne in the documents also refers to her father Frederick as King of Denmark and Norway which as his formal style would have been used. And her brother Ulrik is once referred to as heir of Norway oddly. The kingdom would have been styled the Kingdom of Denmark and Norway formally, in the same way as the English possessions were properly the Kingdom of England, (France nominally) and Ireland. But Denmark was definitely the standard reference. And the conjunction name of Denmark-Norway appears to be a modern invention.

Reference for English sources:

"The Progresses, Processions, and Magnificent Festivities, of King James the First: His Royal Consort, Family, and Court; Collected from Original Manuscripts, Scarce Pamphlets, Corporation Records, Parochial Registers, &c., &c. ... Illustrated with Notes, Historical, Topographical, Biographical and Bibliographical, Volume 1", John Nichols, 1828.