No - National Traffic and Motor Vehicle Safety Act (15 U.S.C. 1381 et seq.) places the responsibility for classifying a particular vehicle in the first instance on the vehicle's manufacturer. The NHTSA follows that model, and while technically they could disagree with a manufacturer and over-rule them for the purposes of safety testing, I don't believe that happens very often if at all.

For the record, from a 'size perspective', the Model X is much bigger in person than it looks in photos, and fits well in to the cohort of vehicles calling themselves SUVs.