The answer is complicated, because the people who immigrated to England in the 5th and 6th centuries came in small groups from all across Europe, and weren't a single, unified people group or army that can be directly compared to the Danes (who, also, weren't a single unified group).
We know a lot about where people came from during the early Anglo-Saxon period by studying their teeth (from excavations of cemeteries). When you're young and your tooth enamel is still forming, the food you eat and the water you drink leave trace elements inside your teeth. These elements contain different concentrations of stable isotopes which vary geographically; so you can analyze a tooth, find the proportions of isotopes of Oxygen or Strontium, and match that to a general geographic area to see (roughly) where a person spent their childhood. This is really useful, because it allows us to trace the migrations of individual people. It's challenged a lot of our old ideas about the Angles, Saxons, and Jutes.
What were these ideas that were challenged? A few written sources from the early middle ages describe invasions of Saxons, etc, in the 5th century. Bede spells it out most clearly: Hengist and Horsa invade in 450 and start conquering everything. But studies over the past 25 years have shown that Bede (who was writing around 730) really didn't have much evidence for the events that took place 300 years before, and is generally a very unreliable source. His primary concern was spinning England's early history in such a way that it could support some debates he was involved in about the correct model for eighth-century church structure. So his history was not only factually unreliable - it was also very political, and has to be read more as an argument for the correct shape of England's history (in Bede's opinion) than an objective reflection of the actual shape of events. See Walter Goffart (1988), and Guy Halsall (2013) for more detailed discussion.
What this means is, the story of big armies of Angles, Saxons, and Jutes invading and conquering Britain isn't well supported in the historical sources.
Enter archaeology. Certain types of graves and artifacts have, for a long time, been interpreted as belonging to the Germanic tribes. Specifically, the argument was that only Angles, Saxons, and Jutes were buried with weapons. Additionally, new styles of art were very similar to art styles in Germany, and these were seen as being evidence of the invasion of people from Saxony, Jutland, etc. But there are problems here, too. The biggest is that there's very little of this material in England - the scale suggests a much smaller, poorer, and less organized migration than the old 'invasion' story Bede told (see Robin Fleming, Britain after Rome, for the most recent overview of this perspective). Furthermore, there's actually no good evidence to suggest that burial with weapons is a 'Germanic' thing - it actually starts in the Roman empire (see the essays in G. Halsall 2010). And the art styles that look so 'German' are actually new styles that only develop in Germany around the same time as they show up in England. So things change in Britain, but the change can't be connected clearly with evidence of a mass-migration and conquest. Archaeology confirms that the old 'Saxon invasion' argument is problematic.
And this is where stable isotope analysis comes in. Analysis of people's teeth has shown that a lot of people did migrate, but they're often not the people we thought were the Angles, Saxons, and Jutes. Many people with Scandinavian jewelry were born in England; many people who looked 'native' were immigrants from the Baltic. There's no clear connection between burials with weapons and 'Germanic' origin. And lots of people came from all over Europe, not just Scandinavia and norther Germany.
So: People migrated into England in the 5th and 6th centuries. But they almost certainly weren't big armies from Saxony or Jutland. They were small family groups from all over Europe (including, but not limited to, Scandinavia and Germany). Many of these people would have shared ancestors with the ancestors of the 'Danes' who invaded England in the 9th century. Some would not have. There was a lot of diversity.
And the same could be said of the Danish army: they, too, were composed of people from all across the North Sea region. They almost certainly would not have shared a single common language or heritage before joining up with the Great Army.
Now, that said, England's language eventually became Old English, which is related to the Norse languages spoken in Scandinavia in the 9th century. There may have been religious similarities between Anglo-Saxon 'paganism' and Norse religions (though it's hard to say, as we have very little evidence from which to interpret early Anglo-Saxon beliefs - all the specific evidence we have for 'Woden' is from after Britain's conversion to Christianity). So we can make some broad comparisons between many of the people in 'Anglo-Saxon' England and the later invaders from Scandinavia. But we have to be cautious not to assume too much similarity, because neither group had a unified culture, language, heritage, or history: both were multi-ethnic, diverse, gathered together from many parts of the world, and only labelled as 'Danes', 'Angles,' 'Saxons,' or 'Jutes' after the fact by people trying to make sense of what exactly each group was. So we can't assume things true of one group (like religion or gender norms) will necessarily be shared by the other.
What is the relationship between the early Danes and the Angles, Saxons and Jutes who inhabited Jutland?
As /u/alriclofgar has noted, the Angles, Saxons, and Jutes didn’t really exist as we think of them today until the time of Bede (early 700s). The people in Britain who started calling themselves Angles, Saxons, and Jutes were probably of mixed heritage: Celtic, Romano-British, Frisian, Frankish, German, and anything else you can think of. But by the 600s, these people started to think of themselves as belonging to regions with distinct identities. Unsurprisingly, these regions were generally defined by waterways (the Thames, the Severn, and the Trent). So the Angles, Saxons, and Jutes were really peoples who created themselves in England, even though they called themselves by old names for people living on the continent. As the archaeologist John Moreland (2000: 35) succinctly states: “Contrary to common understanding, it would be a gross error to assume that the Angles, Saxons and Jutes possessed discrete identities in their so-called tribal homelands and maintained them in the early part of their settlement in England.” Nevertheless, Englishmen like Bede were quite proud of their presumed continental heritage: they wrote a lot about it. Unfortunately for us, most of this was the stuff of legend and tells us nothing reliable about the situation on Jutland, no matter how authentic it sounds.
Regarding the people who actually lived in Jutland, we have little contemporary evidence (outside of our English writers) that they ever considered themselves as Angles, Saxons, Jutes, and they might not even have called themselves Danes until the late 900s. In the 500s, historians such as Jordanes, Procopius, and Gregory of Tours did start referencing Danes, but their information is sparse. Jordanes and Procopius lived on the shores of the Mediterranean, and it’s impossible to judge the veracity of anything they say about the distant north. Gregory of Tours is better situated in central France, but he only tells us that the Danes and Saxons were defeated by the Frankish general Lupus, who “drove them from the Wupper to the Lahn” (III.3). The general impression is that these Danes and Saxons had advanced out of central Germany, and the Franks drove them back south along the Rhine, halfway from Düsseldorf to Frankfurt. So the situation in the 500s is that people using the names Danes and Saxons to describe people they considered to be troublesome barbarians coming out of somewhere in Germany, but their information is neither exact nor reliable. It doesn’t tell us anything about Jutland.
By the 800s, people in England and France called pretty much anyone that spoke Old Norse a Dane, regardless of where that person came from. Meanwhile, the people living in what is today Denmark didn’t have any sense of belonging to a single community. In English and Frankish sources, we start finding reference to the “North Danes” and the “South Danes,” which suggests a politically divided linguistic area rather than a single ethnic community. It seems that the first person to coin the term “Denmark” was the Englishman Orosius in the 890s as he struggled to make sense of this complex situation. But in this case “Denmark” really just means “borderlands of the Danes,” i.e. the place nearest to England where a lot of people speak Old Norse. And if you were to follow the currents of the North Sea from the shores of England, this would indeed take you to the shores of modern Denmark. This is an important moment, because it’s the first time that anyone started thinking about Denmark in a way that could describe a single unified community, but Orosius was an outsider and he was talking about a linguistic area and not a political state.
To my knowledge, the first indication that at least some people living in today’s Denmark were really thinking of themselves as Danes is the Jelling Stones. Harald Bluetooth, who was a Norse-speaking king, set up these runestones around 970 to claim that he “won for himself all of Denmark and Norway and made the Danes Christian.” So in this instance, it looks like Denmark and Norway are places, and Danes are the people who Harold rules and who speak Old Norse, regardless of where they lived. This included Norse-speaking people in Normandy and along the south Baltic shore, in addition to the aforementioned people in Denmark and Norway. It was a bold claim, but it’s important to note that at least some semblance of political unity was needed before all these Old Norse speakers started thinking of themselves as a single community of Danes.
Only in the nineteenth century did the modern story of Angles, Saxons, Jutes, and Danes emerge. Early historians assumed that their medieval predecessors had preserved authentic information, and they didn’t worry about bias or purpose or even whether the information was reliable. They also lived in a time when nation-states were on the rise, so they assumed that any reference to a people was based on an ethno-political identity, which historians now recognize was tremendously anachronistic. This is still a strong (and popular) narrative, but the evidence as we have it gives us little reason to think that anyone living on Jutland ever considered themselves Angles, Saxons, or Jutes during the early middle ages, and only after Anglo-Saxons in England and Franks in Gaul and Germany had been calling Norse speakers Danes for centuries did the people united by Harald Bluetooth and his successors begin to call themselves Danes. People in Jutland may have begun calling themselves Jutes at about this same time, as missionaries from England were coming over under the assumption that they were already called Jutes. The name “Jutland” may have came even later, maybe even as a bid for regional identity against the encroaching “Danish” kings. But this last bit is only speculation.
So in short, we have no evidence that people living on Jutland ever considered themselves Angles, Saxons, or Jutes. Their North Sea neighbors called them Danes throughout the early middle ages (along with anyone else who happened to speak Norse), but the people living in modern Jutland only started thinking of themselves as Danes when strong rulers started enforcing their authority over modern Denmark and southern Sweden as part of a unified kingdom. Only later did people start rationalizing the early medieval sources and begin to reconstruct Angle, Saxon, and Jutish identities for the ancestors of people living on today’s Jutland.
A few good secondary sources:
Clare Downham, "Viking Ethnicities: A Historiographic Overview," History Compass 10.1 (2012): 1-12.
John Moreland, “Ethnicity, Power and the English,” in William O. Frazer and Andrew Tyrrell (eds.), Social Identity in Early Medieval Britain (London: Leicester University Press, 2000), 23-51.
