The descriptor predates the the automobile by quite a bit.
Without any context, it seems weird that "spider" refers to an open-top, two-seater car, and the term shows up throughout the auto industry on vehicles like this from the affordable Fiat 124 Spider to the Porsche 918 Spyder supercar. A new video from Carfection explains how the arachnid's name relates to roadsters.
The term actually predates cars by many years and originally comes from a type of horse-drawn carriage. The term phaeton is likely familiar from Volkswagen's defunct flagship sedan, but the word comes from the phaeton carriage that features a high-sitting, lightly sprung body, open seats, and large wheels. The phaeton spider is a sub-type with an even more compact body but retaining the big wheels. According to Britain's Heartland Carriage, which still builds them, the compact, simple design allows for easier maneuverability and puts the visual emphasis on the owner's horse. The original craftsman apparently thought the layout resembled a spider because of the spindly body and the multi-spoke wheels of the time were like a spider's multiple legs.
When the automobile arrived, many of the coachbuilders that built carriages switched to making bodies for cars, and the constructors applied the old vocabulary to the new vehicles. When describing a lightweight, open-roof model with an emphasis on handling and the powerplant, it's not hard to see why these folks would use "spider" to them.
In addition to phaeton and spider, a variety of other automotive body styles also take their names from carriage designs, including coupe, cabriolet, shooting brake, and volante. The modern auto industry owes a lot to the terminology from well before the first car hit the road.
Source: Carfection via YouTube