weird metaphor

You're a better googler than I, Gunga Din. I just tried again with "cover like a rug" and only got 2 hits! I couldn't remember what I used last time to get 5.

I just remembered -- I used the expression from the quote "on this like a rug".

I don't know why I immediately went to "covered like a rug." OK, I just looked at my search history. I didn't go to "covered" immediately.

I started with "lie like a rug."
Then went to "all over this like"
Then "over this like a cheap suit"
Then "covered like a cheap rug"
Then "covered like a rug."

I assume I got a hit in each search that caused me to revise the strategy.

Does 'rug' mean wig in US English as it does in British English, giving the sense of covering something very tightly, even more tightly than a floor covering?

Does 'rug' mean wig in US English as it does in British English

Yes, but it's rather old fashioned, although I found some recent uses, including a story about a "The Toupee Titan of New York" published a couple of days ago, where the toupee business is called the "rug trade."

But I think "rug" is a man's toupee, which is not usually a full wig; a woman's wig wouldn't be a rug. So a rug isn't much of a tight covering.