Until the 1500s or 1600s, Europeans didn't make much distinction between people living in modern Germany or people who lived in the Netherlands. Before 1500, there had never been a country officially called "Germany" or the "Netherlands" yet. That whole area of northern Europe was populated by the "Dutch". The German word for that is "Deutsch" and the Dutch word for that is "Deitsch".

By the 1500s, the Dutch languages had drifted apart enough that the people themselves, and their neighbors like the English, began to make a distinction. The Dutch speakers in the highlands, up the Rhine river, were said to speak "High Dutch", and the Dutch speakers in the lowlands, or "netherlands", down river at the mouth of the Rhine, were said to speak, "Low Dutch".

At this time, speakers of the High Dutch languages largely lived in what was left of the Holy Roman Empire. The Holy Roman Empire in the 1500s no longer had a stake in its namesake Rome, so to placate a number of the princes and dukes and lords who did remain part of the Empire, at the Diet of Cologne in 1512, the empire changed its name. But being the Catholic empire and all, the name had to be in Latin. And Latin, being a dead language, didn't have a word for "Dutch" so they resurrected a long-dormant word, "German", which was a word used in Julius Caesar's time to describe an unrelated tribe of people who at that time lived in the area.

Thus, the Holy Roman Empire became "the Holy Roman Empire of German Nations" in 1512, and in the next decade, the words "German" and "Germany" to describe a language and a place were first used in English writing.

The terms remained rare, however, as "Dutch" and "High Dutch" continued to be the preferred terms for many more decades. But then, in the late 1500s, the Low Dutch speakers on the coast won their freedom from occupying Spain and formed a republic. Their own name for this new country translated to "the Republic of the Seven United Netherlands" but the colloquial English name for it became "the Dutch Republic".

And so, from the early 1500s until the mid-1700s, the word "Dutch" underwent a transformation, following these new political realities. At the beginning of that period, for an English speaker to say the word "Dutch", you could be talking about someone who spoke the local language of Amsterdam or Berlin. If you needed to make a distinction, you'd differntiate by calling them "Low Dutch" or "High Dutch". But by the end of that period, for an English speaker to call someone from Berlin "Dutch" was unheard of, and even to call them "High Dutch" would not be readily understood. It would be something that maybe your grandparents used to say.

By the late 1700s, the word "Dutch" was pretty much exclusively used in the English language to refer to the people and language of the Netherlands who lived in the "Dutch Republic", while "German" and "Germany" came to refer to what used to be called the "Dutch" languages spoken in what was then part of the Holy Roman Empire, and what is now known by the locals as Deutschland.

Meanwhile, in the German and Dutch populations themselves, a similar transition happened at roughly the same time, only instead of the Germans who stopped being called the "Dutch", it was the Netherlanders. So while both German and Dutch people would have called themselves "Dutch" in the 1300s and 1400s, by the late 1600s, the people of the Netherlands had stopped calling themselves that, and instead started calling themselves "Nederlands" and their language "Nederlands", since their country was called "Nederlands". The Germans continued to use the words they always did, so they became indisputably the "Deutsch" who lived in "Deutschland".

Other languages have different origin stories, but they mostly stem from the period from 1500 to the early 1700s when the German and Dutch cultural identities emerged as separate, and there was formed a political entity called the "Netherlands" for the first time, and a place adjacent to it called "Germany". Versions of the word "Allemagne" instead of Germany/Deutschland are used in several languages because, out of all the Dutch languages, it was the Allemanic version that came to be the standard "High Dutch" or "German" spoken. It came to be spoken throughout many of the German nations and, ultimately, it became the version of the German language adopted by various German royals in court. "Allemagne" was originally a small Dutch/German speaking nation on the French border, but it followed its language around so that in French, Spanish, and many other languages, Germany/Deutschland is called "Allemagne" or some variant. "Dutch" usually isn't used in those languages at all and the people from the Netherlands are either referred to as Hollandish or Netherlandic.

And just to address Danish in your original question, "Tyskland" is the Danish rendering of the word "Deutschland". "Tysk" is the Danish word for "Dutch"/"Deutsch". The people of the Netherlands are referred to in Danish as "Hollandsk" people who live in "Holland".