2 copies of a movie can exist simultaneously and both still have value. The bitcoin network exists so that only one copy of a coin can have the value. They are fundamentally different things, even if their physical instantiation happens to be "just bits".
The problem with your analogy is that once spent, your bits become worthless. If for instance, I made a song and then you pirated it and then I no longer had it on my hard drive any more, then that is what's happening here.
Similarly, my password to my bank account is "just bits on a computer", but you could steal that from me, and take all of my [real] money. In that sense, it is theft. Piracy is copying, theft is moving.
If I copy your file and the file is the asset then I don't deny you free enjoyment of that file. If you own the right to prevent me copying that file and are not disclaiming it then an unlawful copyright infringement has occurred. You and I both get to use the file, this is essentially why the file is not "stolen".
If I copy your file and the file acts as a token for a real sum of money, and I acquire that sum of money (be it virtual or whatever), then I deny you free enjoyment of that file. Once used to extract the value it tokenises the file no longer has value and so I deny you the value inherent in the file by my use of it. This is stealing. It may also be a copyright infringement in this case.
Compare the second action to using a one-use credit card number. If you copy the details off my one-use credit card number then you've more than likely acted unlawfully. If you use your copy to spend the money that those details represent then you've stolen from me.
Private keys and such are not infinitely reproducible without cost to their value. Cultural value doesn't degrade over popularity/distribution. Apples and oranges.
I see where you are coming from, however I would bet that the majority of the money you have, with the exception of the coins in your pocket, is already in the form of bits on a computer.
Just like money Bitcoins are numbers in a database. With software I can make 100 trillion copy's and cost you nothing, I can't do that with money or Bitcoins.
Ignoring for a moment the objections of value (which a bunch of other commenters have already pointed out), your two scenarios don't line up at all:
Piracy: A has bits X. A wants to give B copy of X. B wants copy of X. Third party C (record label) objects. [A pirates a copy to/with B]
Bitcoin: A has bits X. A does not want to give B copy of X. B makes one anyway. A objects. Third party C (person who mined coins originally and willingly transferred to A) does not care. [B steals coins from A]
The two scenarios do not line up at all. In fact, if you made a bitcoin scenario that did line up:
Bitcoin': A has wallet X. A wants to give copy of wallet X to B. B wants copy of wallet X. Third party C (original miner of coins) objects. [A wants B to be able to spend his coins for whatever reason]
well, in the Bitcoin' scenario, I think almost everyone would agree that C has no moral authority to object.
Eh, not really. Here's a very clear example of why your stance doesn't work.
I buy an MP3 from Amazon. I then copy it 'copy bits' over to another computer. Agreed, at this point all is fine. However, then I go try to sell the copied song for value to a 3rd-party. I may profess to 'own' the original download and the copy, however that doesn't give me the right to sell the song. It's the same with Bitcoin, if you just copied my Bitcoins, it's not theft in a sense, it's when you try to sell or exchange my Bitcoins for value that theft comes into play.
Bitcoin are just bits on a computer.
Just like piracy, when we copy bits from one place to another, is it really "stealing"?
Anyone that says "yes", better double-check their stance on software piracy as well.