Escalator was once a brand name, owned as firmly as any logo. It lost that status because people used it too well.
Moving stairs had several inventors before they had this name. Wikipedia credits Nathan Ames with an 1859 patent for "revolving stairs" that was never built. Jesse Reno patented an "Endless Conveyor or Elevator" in 1892 and installed one beside the Old Iron Pier at Coney Island in 1896. It was a sloping belt, not a set of steps.
The name came from Charles Seeberger, who bought another inventor's patents and built flat, moving stairs. He trademarked "escalator" in 1900. By Wikipedia's account, Seeberger consulted a Latin lexicon and took scala as the root, with "E" as a prefix and "tor" as a suffix. Etymonline gives a different breakdown: escalade, an old word for scaling walls with ladders, plus the -ator of elevator. Either way, the core is the Latin for ladder or steps.
Seeberger had partnered with the Otis Elevator Company in 1899. Their machine won first prize at the 1900 Paris Exposition, and Otis held the trademark. Rival firms had to use other names, such as Motorstair, Electric Stairway and Moving Stairs.
None of them caught on. The public called every moving staircase an escalator. Merriam-Webster says Otis "did little to protect its rights to the mark." In 1950 an Assistant Commissioner of Patents ruled in Haughton Elevator Co. v. Seeberger. The ruling said the public saw the word as "the name for a moving stairway and not the source thereof." That made it generic, free for any maker to use.
The word then kept climbing. Etymonline says the verb "escalate" was formed from escalator in 1922, meaning "to use an escalator." By 1959 it had taken on its now-familiar sense of raising, first in scenarios for possible nuclear war. A brand name for a staircase became the standard word for things getting worse.