If I buy, say, a share in The Walt Disney Company, does that give me part ownership in the intellectual property?
Does this ownership then give me the right to copy it?
No. Only the company owns the copyright, not its shareholders. A company is a separate legal entity with its own capacity to own property. Copyright is property, not infection that can be spread onto whoever is close enough. Pretty much like shareholders can't just share the use of the company's tangible assets, they can't wet their beaks in the copyright.
As a shareholder, what you could do is to appoint a director/CEO that will give you license to copy (or even transfer the copyright) — if you have enough shares to fire the current CEO and put yours. The CEO will get into legal trouble with other shareholders/creditors, but you'll still enjoy your right to copy.