No, The Doors is not a goth band, because they play Psychedelic rock, while goth music consists of Gothic Rock and its derived genres, like Deathrock, Darkwave, Ethereal Wave, and Coldwave.

However, they were an important influence on the development of goth music and are respected in the goth scene.

If you want to dive into the goth music scene, some of the bands that might appeal to you if you enjoy The Doors are Bauhaus, Siouxsie And The Banshees, and The Southern Death Cult.