The differences between Danish, Swedish, and Norwegian are much closer than the major Romance languages today are to each other. The Nordic countries never really fell under a single crown, but I could say the same about Germany, Austria, and Switzerland--and the unification of Germany in the 1800s came about long after High German got established as a written standard. Why didn't Low German establish its own written standard that took root in Northern Europe, despite its historical prominence?
I'm pulling a lot of these off from what I remember from my university lectures, so bear with me if it's a little disjointed. I'll try and pull some sources from the slides as well, if you want, however most of them are going to be in German or Swedish.
First off, the Nordic languages aren't Low German (Low German is its own language and much closer to Dutch than to Danish) and they fell under a single crown multiple times. In fact, having three proper Scandinavian kingdoms and languages is a product of the 20th and 19th centuries. The Kalmar Union unified Denmark, Norway and Sweden under one crown from 1397 to 1523 and even after that Norway was Danish until 1815, when it nominally became it's own country but was still under a Swedish king.
Do note that all three actually did use to be dialects of the same language. Old Norse is usually considered to be split into West Norse and East Norse, with Iceland and Norway speaking West (more on Norway later) and Sweden and Denmark East Norse. Older texts will sometimes talk about something like Rune Swedish/Danish (e.g."Runsvenska"), but those two for example were so close to each other that it would have been impossible to call them separate languages.
The difference between East and West Nordic languages still exists, but at least in the case of Norwegian are they nigh impossible to tell apart by a non-speaker, as it's usually minor grammatical differences ( e.g. slightly different plural endings and differentiation between male and female gender) and pronunciation such as preserved diphthongs. The reason for that is that Norway had been under Danish (and Swedish after 1815) control for more than 500 years, only really starting to consider Norwegian as it's own thing during the Age of Romanticism, and even then writing it with Danish standards, as Danish had been the formal and prestige language before that. Language policy in Norway is still really weird, with there being two different standards to write Norwegian (Nynorsk and Bokmål), one based on mostly western Norwegian dialects, the other being basically Danish, but modified to better fit the local pronunciation (this process happened roughly throughout the 19th an 20th century). Do note that Icelandic and Faroese however have remained fairly untouched and can be considered more or less "pure" West Nordic languages and would have been also much closer to medieval Norwegian than what is spoken in Norway today. If one compares those two to Norwegian one can see the influence Danish (and to a much lesser extent, Swedish) has had on the modern Norwegian language.
Now, onto why Danish and Swedish are different, even though they were pretty much the same language before 1000. Outside of pronunciation (Danish is really quite something else to the point that the often-claimed mutual intelligibility isn't quite what it is made up to be) it really comes down to developments in language, politics (and, in a sense, religion). While the spoken language already differed in late Medieval times, their written forms were still fairly similar. What changed the game was print and the reformation. Both Denmark and Sweden became staunchly Lutheran and began printing their own bible translations. In fact, the translation of the bible into Swedish has been considered so influentual for Swedish spelling that it's often used as the point where Old Swedish ends and modern Swedish begins ("Fornsvenska" to "(Äldre) Nysvenska").
Take as an example the lord's prayer, first Swedish, from the bible of Gustav Vasa (around 1541), and then in Danish from the late 15th century (both slightly reformatted).
Swedish:
Danish:
(The bible translation from 1550 would probably have been a better comparison, but I didn't manage to find the text on short notice.)
You'll note that Danish already uses æ and ø and Swedish already has ö, ä and å (which didn't enter formal use in Danish until the last language reform in 1948). Additionally, a lot of words that use a hard consonant in Swedish, for example "rike[t])" (for comparison: ON: "ríki"), have already adapted to the shift to softer pronunciation in Danish, becoming "righæ" (modern Danish: "rige"), or already soft consonants changing into vocals ("daghligha"/"daulighæ", modern "dagliga"/"daglige", do note that the modern Danish variants is still pronounced as if it's written with au). To conclude, both languages adapted their spelling to the local pronunciation of words, with the Swedish of the period being more conservative than Danish. Interestingly enough, these days Danish orthography is extremely conservative in comparison to its spelling due to the rapid development the language underwent in the last 100-200 years while Swedish remained relatively stable and thus has closer links between spelling and pronunciation. Norwegian however is by far the closest in that regard, as both variants in spelling have underwent multiple reforms throughout the last two centuries.
Both Swedish and Danish attempted to standardise spelling much earlier than your given example, German, mostly because unlike the German "Flickenteppich", they were centralised states with the ability to do so. The Swedish Academy was founded in 1786 with the specific goal of providing standards for Swedish spelling and grammar (published 1801), while the Danish government published a set of rules for written Danish to be used in schools in 1775 (do note that there were already inofficial standards in place in both countries). German, on the other hand, while Standard German did already exist in some form, didn't receive a formally unified set of rules until after the unification (first semi-failed attempt in 1876, then 1901) and even then every part of Germany had to enact them separately and even then it's only considered binding in certain contexts, even today.
Interestingly, in 1869 there were attempts to bring the three languages closer to each other, primarily by reducing loan words (mostly from Low German and French) and (re-)introducing native words, some of which still exist ("idrott" instead of "sport"). However, when it came to more unified spelling, the proposed reforms were largely ignored by the authorities - the Swedish Academy for example only formally introduced the change from f to v in words like "av" almost half a century later in 1906, something that had been done significantly earlier in, for example, the then emerging Norwegian. Other than that, they continued based on the original 1801 proposal.
Long story short, there were two major factors for how spelling works in the three countries - first there is pronunciation (seriously, Danish is bloody weird), then there is tradition. Changing the way people are supposed to spell is something that always causes enmity within the population, to the point where it was and still is a point of contention in Norway, especially last century and even today people from majority-Bokmål areas (which is by far the larger part of the country) have to learn Nynorsk (or "spynorsk", ""barf-wegian", if you ask certain pupils).