According to the same Wikipedia article you cite,
...the zero point is determined by placing the thermometer in brine: he used a mixture of ice, water, and ammonium chloride, a salt, at a 1:1:1 ratio. This is a frigorific mixture which stabilizes its temperature automatically: that stable temperature was defined as 0 °F (−17.78 °C). The second point, at 32 degrees, was a mixture of ice and water without the ammonium chloride at a 1:1 ratio. The third point, 96 degrees, was approximately the human body temperature, then called "blood-heat"
According to a letter Fahrenheit wrote to his friend Herman Boerhaave, his scale was built on the work of Ole Rømer, whom he had met earlier. In Rømer's scale, brine freezes at zero, water freezes and melts at 7.5 degrees, body temperature is 22.5, and water boils at 60 degrees. Fahrenheit multiplied each value by four in order to eliminate fractions and increase the granularity of the scale.
Rømer's choice of 60$^\circ$ as the boiling point of water makes sense if you consider the fact that Rømer was an astronomer, so 60 has special significance. So really it was Rømer who pioneered non-decimal-based temperature scales, Fahrenheit was just following suit.