No, Old Moon is not a goth band, because they play Post-Punk/Dream Pop/Indie Rock, while goth music consists of Gothic Rock and its derived genres, like Deathrock, Darkwave, Ethereal Wave, and Coldwave.

However, their music does show some goth influences and could be considered adjacent to 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 Old Moon are Fearing, Soft Kill, and Tragic Figures.