Its pretty obvious virtually all flames are the energetic reaction between a fuel and an oxidizer. You understand rightly that both fuel and oxidizer are consumed, joined to create something else. But since we live in an oxygen background, oxidizer is usually naturally present. Fire then becomes a likelihood only when fuel is added to your situation. So we started saying "flammable gas" for a strong or hi-energy fuel gas. Oxygen is a strong or hi-energy oxidizer, true, but it is an oxidizer, so it is not a fuel, so they do not call it flammable.