Music copyrights are property, but they aren’t property like any other type of property.
Basically, ASCAP and BMI license 99.9% of the available copyrighted music in blanket performance licenses. The price for these differ if you are a catering hall, a bar, a dance studio, a nightclub, a radio station, or a stadium (or, for that matter, a travelling political campaign), but if you have the licenses you can pretty much play any music available in any medium over the PA system within the scope of your license type without committing a copyright violation. I’m not positive what the situation is with online play – at one point, Congress was creating a blanket license and license fee, but the rights-management companies (ASCAP & BMI) may be involved now.
The regular performance license also permits someone to come in with a guitar or a piano (or a cover band) and perform a version of a song written by someone else (or even by him- or herself, if someone else holds the copyright, which is usually the case).
If you want to record a song, and you don’t personally own the copyright, you need to get a different sort of license. If you want to alter the song in some significant way, you need specific permission for that. If you want to sample someone else’s recording of a song, you need a completely different license, often from a different company altogether – the record company, vs. the publisher or a rights-management company. And if you want to incorporate someone else’s recording or a copyrighted song into a film or TV ad or internet clip, you need yet another kind of license. All of these sorts of licenses are negotiated more on a case-by-case basis, which lets some artists, say, keep their music out of commercials (and other artists license their songs to commercials for cheap in order to get exposure). Sometimes it gets very hairy – there was an article in the New York Times this weekend about a young indie artist who had to recall and trash every physical copy of his new album because Ric Ocasek never agreed to a license to sample “You’re Just What I Needed”. And the DVD of Cameron Crowe’s movie Almost Famous included a sequence that was dropped from the movie because they couldn’t license “Stairway To Heaven” for it. On the DVD, it was provided without a soundtrack, but with instructions when to start playing the song on a separate device.
In general, nothing in copyright law or the terms of standard performance licenses would stop a candidate from playing a particular song at a political rally. If the venue and/or the campaign have an appropriate blanket performance license, what they are doing is perfectly fine from a copyright perspective. (Not true if they are putting footage of the rally and the song into campaign ads, though.) But, as the ASCAP sheet points out, there is non-copyright law – law that is much harder to enforce, by the way – that attempts to prevent one person from giving a false impression that another has endorsed the first person, or from otherwise profiting from an unlicensed use of the other person’s image and reputation.