Does unscented mean fragrance-free?

Not always. An unscented product can contain ingredients that mask its smell. For someone with fragrance sensitivity, the exact label matters because masking fragrance can still be present in an 'unscented' formula.

Unscented and fragrance-free are not the same thing

If fragrance is a concern, look for fragrance-free labeling, while remembering that this still does not guarantee that every other ingredient will suit your skin.