The reason for this has to do with the history of Germany and the Netherlands and northwestern Europe in general.

Until the 1500s, there was no unified political place called "Germany", nor was there a place called the "Netherlands" either. The people in both places were referred to as "Dutch" or "Deutsch"--take your pick. It's the same word, or at least it was back then. If you were an Englishman living in 1400 and met somebody from Amsterdam or Berlin, you'd call them "Dutch" and the language they spoke "Dutch". You didn't make much of a distinction, because "Dutch" encompassed a series of languages, not just one. Not only did it include the precursors to the modern Dutch ("Nederlands") and modern German ("Deutsch") languages, but also other still-living variants and cousins such as Flemish, Frisian, Saxon, Limburgish, Zeelandic, Brabantian, Swiss German, and others. And this is going back aways, when English speakers were still speaking Middle English. So, too, were these Dutch/German variants speaking earlier versions of themselves.

By the 1400s, there had arisen enough of a difference between the dialect groups that English speakers began making a distinction. The Dutch people who lived in the lowlands (or "netherlands") at the mouth of the Rhine river were said to speak "Low Dutch" while those who lived upriver, in the highlands, were said to speak "High Dutch" (which became modern German).

In the 1500s, a couple of political events effected the vernacular. First, the Holy Roman Empire, at the Diet of Cologne in 1512, formally adopted a new name. The remaining political stronghold of the empire had nothing to do with Rome anymore. It was mostly in northern Europe. But, this being the Catholic empire, the official language continued to be Latin. And since there was no direct Latin word for "Deutsch" or "Dutch", being a dead language and all, they renamed themselves "Imperium Romanum Sacrum Nationis Germanicæ", or "the Holy Roman Empire of German Nations". "German" was a dead word from Julius Caesar's day that referred to an unrelated tribe of people who lived in that area back then.

Shortly after the Diet of 1512, in 1520, the word "German" first entered the English language as a description of a nationality, though "Germany" had already come into colloquial use through the Catholic church before then.

But it was the formation of the Netherlands that really cemented the distinction. In 1549, the Seventeen Provinces in the Low Countries ganged up to force French and Spanish occupying forces out, leading to the establishment of the "Republic of the Seven United Netherlands". In Hollandic Dutch, they called it "Republiek der Zeven Verenigde Nederlanden" but coloquially, in English, it became known as simply the "Dutch Republic".

And so, from that point, the terms "German" and "Dutch" began to refer to different things, but it took some time. Over the course of the 1600s and into the early 1700s, an English speaker could speak about the "Dutch" or something being "Dutch" and it could be understood to mean either the Netherlands, Germany, or some other land or language in that general area of Europe.

By the mid-1700s, "German" and "Germany" had come into common usage as a synonym for "High Dutch" and the "Holy Roman Empire", while "Dutch" increasingly meant the Netherlands specifically. By 1800, using "Dutch" in any way outside of referring to the Netherlands had become rare.

Long story short, "Dutch"/"Deutsch" used to refer to both people from Germany and the Netherlands, in not only English, but in German, and in Dutch as well.

Due to some political events in the 1400s-1600s and increasingly divergent languages, "Dutch" in English came to refer only to people in the Netherlands, while "Dutch" in both German and Dutch came to refer only to people in Germany.