Question: Is white vinegar, aka the stuff just labelled as "vinegar" in the US and which I use for cleaning my kettle, the same as "White wine vinegar", which I have purchased on accident a time or two? If not, what is white vinegar made of?

Answer: 
No, it's not. White vinegar (also known as distilled vinegar) is made of acetic acid diluted in distilled water. Its flavor is simple—its just sour. Typical concentrations range from 5–7% acetic acid.

White wine vinegar is made by allowing white wine to turn to vinegar. It has a much more complex flavor profile. It is also frequently less sour (acidic) than distilled white vinegar.
White wine vinegar is a completely different thing, it's less tangy and is more diluted as it's made from the white wine. The Ethanol in the white wine is let to oxidise into ethanoic acid, which is a carbolyxic acid also known as vinegar.