There is a very substantial difference: Copyright infringement does not deprive the creator of the original work, unlike theft which does.
Historically this difference is highly noticeable: Property rights to physical items and land have existed pretty much as long as humanity to various extents, even in societies with no formal concepts of jurisprudence and no centralised enforcement. It has been codified in laws pretty much as far back as we have records - the first codified property laws are more than 4000 years old.
While copyright came into being as an Act of Parliament with the Statute of Anne in 1710, which was an explicit restriction of rights, and it took a further 150 years or so to solidify the modern copyright system, because it was essentially all invented from scratch with no basis in previous human history. Most of these efforts explicitly recognised copyright as a bargain where the public gave up rights, while property law has systematically been justified with the idea of a natural right to control that which you can physically possess and protect.
Incidentally I believe property rights are too extensive too, and e.g. in Norway where I'm from there are substantial ancient exceptions from private property rights (on the basis that too extensive property rights also involves the public surrendering freedoms) - the "freedom to roam" ("allemannsretten"), but historically they are still of a fundamentally different character in that at least the underlying concept of property law basically devolves to the idea that someone does not have the right - though the may have the power - to deprive you of access to that which you physically possess.
> And the way to handle this is to lobby to get the right redefined to have the sort of scope you want.
Meanwhile most people are not interested in a crusade - they just want to go on with their lives, and carry out actions they largely don't see as morally wrong. History is full of laws that have fallen because enforcement became impossible due to total public disregard for the law.
> The "abused their position" canard is becoming toothless in the face of how much money is on the opposition side now
As long as the copyright terms are as long as they are, the DMCA is still on the books in the US, and sites like Youtube enforce far stricter standards than they are legally obliged to, it is by no means toothless.
> The lobbying dollars are there to weaken copyright protections to give these distributors a better negotiating position.
Of your two examples, one (Apple) has a strong self-serving interest in protecting copyright, and the other is a site run by a company with a strong self-serving interest in protecting copyright, thanks to iTunes and the Play store respectively. I don't see why you believe that either Apple, Google or similar companies have much to win by lobbying for weaker copyright protections. Nor have I seen any kind of evidence that they do.
> It's a hard political sell because in the U.S. we just believe in really strong protections for property rights.
Copyright is not a property right, no matter how many times they repeat the term "intellectual property". Copyright law exists in the first place exactly because property law fundamentally is based on the concept of possession. Copying does not remove possession of anything, and so it could not be a violation of property rights.
And the extent of piracy in the US too demonstrates quite well that whether or not people think it is "right" in abstract terms, a substantial proportion of the population does not consider it wrong enough to stop participating in it. Copying and sharing has been an ingrained part of human culture since before written history. In fact, large parts of human culture throughout history has depended on widespread copying and sharing.
So on hand, the historical state has been one of freely copying and sharing, while on the other hand the historical state has been one of rapid and continuous strengthening of enforcement of physical property rights. The two are about as opposite as they can get.
> We like our property rights absolute
Not even the US has property rights that are anywhere near absolute. Try erecting a ten story building on the plot of your suburban house without applying for planning permissions and fighting all your neighbours. Try buying a house right in the path of a long planned road and fight the application of eminent domain. Try starting a brothel most places. Or dumping hazardous chemicals. Or refusing black people access to your store.
Many US states further have exceptions that limit your ability to even prevent access to your property in certain situations - e.g. you may not be able to prevent access to the shore-line.
Your freedoms and rights to do things with your property are continuously weighed against and limited by the concerns for the freedoms and rights of others. Your property ownership does not exempt you from business regulations, or anti-discrimination laws or environmental regulations or a whole host of other laws and regulations that limits what you can do to or on your property.
And that is the one thing that brings property rights closest to copyright laws: They both reflect tradeoffs between the freedom and rights of the public and freedom and rights of the owner.