Copyright law is a property right.
Only in the sense that it says where you cant exercise your rights in your property.
The sense in which copyrights are a form of property is that they can be bought and sold on the free market, can appear as an asset on the books of a corporation or individual, and so on. However, the property is the "copyright" itself, not the content that it protects.
In this limited sense, it's not wholly different from real estate. At common law, you cannot buy or sell land. All land is owned by the monarch and she does not make it available for sale. However, the monarch does graciously allow her subjects to use her land from time to time, and to that end, she issues deserving subjects a form of licence to use the land according to her terms: such a licence is known as an "estate" in the land, and instead of owning the land itself, the licencee only owns title to an estate in the land. Traditionally, estates came in various flavours and could have various conditions or expiration dates (commonly the death of the original licensee), but these days there are only a couple forms in common use.
Even though the United States has never had a monarch, it substantially adopted the English system of land ownership, and still uses the same terminology.
The analogy here is that in some sense you cannot own land, but you can own a privilege to exercise a set of rights relative to that land, and that package of rights is something you can buy and sell. In other words, it's the package of rights relative to the land that is the "property" (not the land itself). Similarly, with copyright, the text of a book, or the creative content of a painting, or a software program, are not property, but the "copyright" itself is in a certain sense a form of property.
This isn't a defence of copyright in any way. It's just useful to understand fully what we are up against.
...You can copyright software...
Since this thread has become ground zero for highly technical points, I should note that copyright isn't something you
do or apply for, as the language "you can copyright" might suggest. Rather, copyright is something that
subsists in certain forms of creative works, from the moment the creative work comes into existence, subject to certain technical conditions:
17 USC § 102(a).
In other words, you don't copyright books, software, or art: the copyright
subsists as soon as you write the book, implement the software, or produce the art.
The question isn't whether "you can copyright" a given form of creative content; rather, the question is whether copyright subsists in that creative content.
In this respect, copyright is far different from patents, insofar as 35 USC § 101 provides that inventors "may obtain" a patent by making an application through the procedure described in the legislation.