Does ginger ale have the same effect as taking a ginger supplement?

Yes. It is well known that ginger has anti-nausea properties --- ginger crackers, ginger chews, and even raw ginger can soothe a nauseated tummy. Ginger ale does have ginger in it, also, but it is wise to drink it slowly and at room temperature.
For most people, yes. If using ginger ale for nausea, regular (not diet) better tolerated, at/near room temperature, and allowed to go part-way "flat.".