“We boil the potatoes and pretty well cook the life out of them.”

This statement admits the nutrients are destroyed.

Boiling destroys many heat-sensitive nutrients especially Vitamin C and B

"When you use the soap on your skin, you receive the benefits of B and C Vitamins, various minerals, carotenoids and flavonoids."

This is not scientifically or regulatory-wise true. Why it is inaccurate?

Vitamin B and C are heat-labile and destroyed during boiling!

Carotenoids and flavonoids are destroyed or reduced by cooking!

Saponification destroys or denatures most plant nutrients!

Even if trace amounts survive, soap is a wash-off product, meaning there is no absorption.

"Basically, it's outer nutrition for your skin!"

This statement is not true and not compliant. Soap does not deliver nutrition. "Nutrition for the skin" implies physiological improvement which is not permitted for cosmetics.

Bottom Line (plain and direct)

By the Sea Soap Shoppe is effectively telling customers that using their potato soap delivers vitamins and nutrients into the body through the skin. 

• The claim is scientifically false
• It is not allowed for cosmetic soap
• It exposes both the manufacturer and retailers to misleading advertising risk