Wikipedia claims that it dates to 1603, but it doesn't explain how this came to be.
Is Wikipedia correct, and how exactly did the word "piracy" become associated with copyright infringement?
Wikipedia is talking about the word pirate, not piracy. Specifically, I think it's referring to:
Banish these Word-pirates (you sacred mistresses of learning) into the gulfe of Barbarisme.
1603: The wonderfull yeare
This quote is listed in square brackets in the OED, meaning that it is deemed relevant to, but not an example that falls under the definition (which is "A person or company who reproduces or uses the work of another (as a book, recording, computer program, etc.) without authority and esp. in contravention of patent or copyright; a plagiarist.")
The first example that's not in square brackets for "pirate" in this sense is from 1668:
Some dishonest Booksellers, called Land-Pirats, who make it their practise to steal Impressions of other mens Copies.
Brooks' String of Pearls
As for piracy, the OED has a bracketed entry for it too, from 1654. The first actual example it gives is from 1701:
Piracy, Piracy, they cry'd aloud, What made you print my Copy, Sir, says one, You're a meer Knave, 'tis very basely done.
A Journey to Hell: or, a visit paid to the Devil: a poem
Piracy meaning the unauthorized use of another's production, invention, or conception especially in infringement of a copyright draws on the older concept of piracy meaning an act of robbery on the high seas;
Pirate, with the above meaning dated back to the beginning of the 18th century:
Meaning "one who takes another's work without permission" first recorded 1701;
Note that the 1603 citation in Wikipedia is not supported by a citation.
Piratia (robbery at sea) is found in post-classical Latin, in British sources from 1419. (OED)
The first reference to it in extended use is from 1606:
1606 T. Dekker Newes from Hell sig. F4v Go & deliuer my most-most [sic] hartie condemnations to all those that steale subiects hearts from their Soueraignes, say to althose, they shal haue my letters of Mart for their Piracie