It’s not really a mix-up. More a continuation of an old name for the language spoken in the Netherlands. The Dutch centuries ago called their language Diets/Duuts/Duits which means something like Germanic. This was before the countries Germany and the Netherlands existed.

Diets is not a single language but a name for all the different regional languages spoken in the low lands also known as Middle Dutch. The name was used to differentiate the languages from the Romance languages.

Hence why the English called the people of the low lands Dutch since the people of the low lands said they were speakers of Diets/Duuts/Duits.

Also in the Dutch national anthem there is a line that says “Ben ik van Duitsen bloed” “I am of Dutch/Deutsche blood” which does not refer to modern day Deutschland but to what all Germanic people in the low lands, what is now present day Netherlands, would call themselves.