Dutch, not an uncommon nickname, seems to be used for a variety of reasons. It could be because a person is connected to Holland, or because a person is of German descent. Here, "Dutch" is from "Deutsch," the German-language word for "German." Other reasons are more idiosyncratic. Here are some examples:
Ronald Reagan, 40th President of the United States
As a boy, Reagan's father nicknamed his son "Dutch", due to his "fat little Dutchman"-like appearance, and his "Dutchboy" haircut; the nickname stuck with him throughout his youth.
Dutch Schultz, notorious gangster
His old associates dubbed him "Dutch" Schultz in honor of a deceased strong-arm thug who was notorious for dirty fighting at the turn of the century.
Dutch Ruppersberger, U.S. congressman
Dutch is legally part of his name: Charles Albert Dutch Ruppersberger III, but it wasn't always. When Rep. Charles Albert Ruppersberger III (D-Md.) first ran for Congress in 2002, he decided that his 13-letter last name was too long to put on a bumper sticker.
Besides, he says, he needed something catchier, something that would stick in people’s heads. So he opted for his lifelong nickname: Dutch. But there was one hitch: It wasn’t his real name, so he couldn’t put "Dutch" on the ballot.
"When you market yourself, you have to make sure that you have the same name on the ballot, so I needed ‘Dutch’ on the ballot," he said in an interview. "So what I did is I legally -- I’m a lawyer -- I legally added Dutch to my name. So I would go by C -- period -- A -- period -- Dutch, and all of the bumper stickers would say, ‘Go Dutch.’ So legally I added Dutch to my name."
But why did he choose "Dutch"? The Maryland Democrat’s nickname has been --Dutch-- since the moment of his birth.
--When I was born ... the doctor came out and said to my father, ‘You have a big blond Dutchman,’ -- he said. --So they started calling me ‘Dutch,’ and when my mother and he would write letters, he’d ask, ‘How’s the Dutchman doing?’ I’ve been called Dutch all of my life.--
Dutch Fehring, Stanford's winningest baseball coach.
It was during his freshman year in high school that he earned the nickname --Dutch,-- because he was of German descent. After Fehring returned a kickoff 60 yards for a touchdown, a local sportswriter tabbed him --The Flying Dutchman-- and the nickname stuck.
Best Answer
The reasoning for this etymology is as follows: A Dutch door comes in two equal parts; if you split the check with your boy- or girlfriend (I am fairly sure that originally "going Dutch" was not used for large parties, but only for going out on dates), that also comes out to two equal parts.
As John Lawler remarks in the comments, the real etymology is more likely to be derived from the derogatory epithet where Dutch is used to mean thrifty, cheap.