On the demo they have previous visits from your fingerprint listed (so you can get an idea of how common the fingerprint is... at least once enough people have tried the demo). Mine had two visits listed which are not mine when using Safari and private relay. On Firefox there were none (and it tracked in normal and private mode).
On my iPhone, I was surprised to see I had made 20+ visits when in reality that was my first visit to fingerprint.com. It makes me feel better that my fingerprint isn't actually unique!