Many antipyretic and analgesic drugs are also known as NSAIDs for the reason that

You've got it backwards, it should be that NSAIDs have antipyretic and analgesic properties. The latter is much more extensive.

Nonsteroidal Antiinflammatory Drugs (NSAIDs) are a class of anti-inflammatory drugs that do not contain a steroidal structure.NSAIDs have been marketed under more than a hundred or so brand names for more than 100 years, since aspirin was first synthesized in 1898.This class of drugs includes aspirin, acetaminophen, indomethacin, naproxen, naproxen, diclofenac, ibuprofen, nimesulide, rofecoxib, celecoxib, and more. The common western cold medicines on the market such as: White Plus Black, Bactrim, Tylenol, etc. are among the acetaminophen, and the main ingredient of Fen-Phen is ibuprofen.

These drugs have anti-inflammatory, anti-rheumatic, analgesic, antipyretic and anticoagulant effects, and are widely used in the clinic for osteoarthritis, rheumatoid arthritis, a variety of fevers and the relief of various pain symptoms. Although the various drugs may have different compositions and chemical structures, they all exert their antipyretic, analgesic and anti-inflammatory effects by inhibiting the synthesis of prostaglandins.