Does an acetal need to have one hydrogen attached to its carbon?
According to the IUPAC definition, an acetal is any compound with a formula $\ce{R2C(OR')2}$ (R' ≠ H). In the past, "acetal" referred only to compounds of this type, which were derived from aldehydes (one R = H), however, it now broadly encompasses all compounds of this type, including those derived from ketones, which were formerly only referred to as ketals. Interestingly, there was a period of time in which use of the term "ketal" was discouraged (IUPAC says 'abandoned', but many chemists kept using it anyway), but it has now been re-embraced as a term, for the subset of acetals that are derived from ketones.

One last note - the molecule shown in this question is a "spiroacetal" - a type of acetal where the acetal carbon is the only atom joining two rings.