Why do I see circular lines and shapes under a product when lighting it with a speedlight?

The problem is that your speedlight is pretty close to a point source light. You need a large diffuse light source, such as through a diffuser, umbrella reflector(*), or using a light tent.

Pardon my poor highlighting skills with my computer trackpad, but I've outlined:

• the label area lit by the flash in blue;
• its reflected image onto the bottom surface in red;
• the reflection of the cylindrical surface of the rest of the body onto the bottom surface in yellow; and
• the reflection of the shadow of the bottle back onto the reflective label highlighted in the green oval.

Because your lighting is point-like, it creates distinct hard reflection boundaries, as well as hard-edges shadows. That's why the reflection of the shadow back onto the label is so distinct.

Again, the solution is a large diffuse light source. A diffuser lightbox, umbrella, or light tent, will eliminate hard-edged reflections and shadows.

* Note: while an umbrella reflector isn't so much a diffuse light source, its size is many times the size of the object being photographed. The light from a large directional light source like an umbrella reflector will still "wrap around" the small bottle, pretty similar to how a diffuse light source will spread the light around the bottle.