Not just the Dutch.
Smith. Forester. Fisher. Gardner.
Hamilton. Irving. Lennox. Montgomery. Ross. Sinclair.
There is a reason, and somewhat well documented one at that.
In medieval documents, nobility are referred and distinguished from one another, through three means: their given name (e.g. Matthew) and a numeral to denote which Matthew they are in their house (e.g. third), their title (e.g. Duke of Nitra) and their house name (e.g. Csak). So, our Matthew is Matthew III. Csak, duke of Nitra, and that is often shortened to Matthew III. Csak of Nitra (everyone knows that Nitra is seat of a duchy), Matthew Csak (is he is only Matthew Csak around lately), etc etc
Now, once you have commoners, house name goes out of the window, so there's that. They still have given name, and if they are abroad, they are called of Nitra if they come from Nitra, but you can probably see why this would be a problem in Nitra itself...
To solve this, two methods were devised: patronymics/matronymics and descriptive names. Patronymics are basically name of your father appended to your own somehow - our Matthew Csak would be called Matthew, son of Peter, Matthew Peterson, Matthew Petrovich, Matthew Peterfi - depends on the region.
Now, patronymics were used for nobility too if there were no other distinguishing features, and especially for younger people. Just a sidenote.
Matronymics are the same as patronymics, but use mother's name.
Descriptive names solved another problem - it's not that unlikely that there are two Matthews, both with fathers called Peter, especially if the local nobleman is Matthew Peterfi, and you need to tell the two apart. So, you use whatever their distinguishing characteristic is.
This can be anything, from the profession (Potter, Smith), physical condition (Blind, Lame) or just a nickname (Giant).
These names were used in middle ages, and they were used inconsistently, sometimes a guy gained a title (Matthew III. Csak wasn't always Duke of Trencin), sometimes he moved and turned from Matthew Peterfi to Matthew from Nitra, or maybe he got really good at something. As often as not, what they were called depended on the mood of the dude making a legal document.
Once the surnames started to officialize, people took the names they already had, and that's how surnames were born.
As for US citizens, a lot of them americanized their names, so they lost the national words in them (von, de, etc.), this happens even today to a degree, with sometimes hilarious results: a hockey player Miroslav Šatan (the surname has to do with clothes, most likely) dropped the Š for S, and became the devil himself.