Copyright is a red herring - it exists (in the US specifically, and generally, internationally) to protect creative/artistic expression. Short phrases (such as titles of books/songs/movies/etc) cannot be copyrighted, nor can bare facts (e.g., "The Washington Senators won the last game in the World Series by a score of 5-4".)

Further, there are a number of "fair use" exceptions which allow limited use of copyrighted works without the copyright holder's permission.

Trade secret is also not a great match - it protects confidential information/processes used in the conduct of a trade or business which confer a competitive advantage. It's not at all clear to me that the average person using BTC as a medium of exchange is engaged in a trade or business (they might be - but if I buy a cup of coffee, for my own consumption, with BTC, that doesn't sound to me like I'm engaged in a business) or that the use of a particular private key or keypair confers a competitive advantage.

I am not saying there are no property rights in BTC, but I don't think they're going to come from copyright or trade secret.