I’d say a one-person band would just be a person. Unless it’s more like a project with one permanent member who recruits other members on a per project basis. This case looks like the former though.
Speaking of interesting edge cases, there are also one-person bands who pretend to be real groups (like Madlibs various monikers on Yesterdays Universe). I would class those cases as groups because that is clearly the artistic intent. But again Violet Cold doesn’t seem like such a case.