With regard to intellectual property, "ownership" is effectively monopoly because nobody else has the right to produce the good. There might be "inferior" or replacements goods with pop music, but maybe not even that when it comes to things like patents.
However I disagree with your assertion that wealth is income generating property. If I store $5mm under my mattress, I have wealth without generating income: the wealth can be used to purchase a large amount of services over a sustained period of time, but it is not generating more wealth.
The argument about "private property" is not hard to understand, it is just confusing for many people because of the way we use the term in modern language. It is not only to be compared with public property, but also personal property: for example, the apartment I live in, or the comb I use to brush my hair. Interestingly, personal property is almost by definition a form of wealth which does not produce income/accumulate value. Here's a good explanation:https://en.wikipedia.org/wiki/Personal_property#Personal_ver...