No, The Birthday Massacre is not a goth band, because they play Synthpop/Industrial 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 The Birthday Massacre are Claire Voyant, Diva Destruction, and Switchblade Symphony.