I'm not convinced of that. People tend to assume he had Q-like powers, but really, the episode established that he needed technological assistance to do his tricks. He lost his power when Kirk destroyed the machine behind the mirror, and when Trelane came back, he explained that the machine was not the only "instrumentality" at his disposal. So he didn't have godlike magic, he had machinery. And that machinery didn't really do much that couldn't be done with a transporter, a holodeck, and a replicator. He couldn't even create fire with warmth or food with taste. And his observations of Earth were limited by the speed of light, which suggests even less advanced sensors than the Enterprise had.
Perhaps these limitations were only because he was a child? His parents, for all we know, could have had Q-like powers, and Trelane himself will, one day. [edit: I see another commenter had the same thought.]
To restate what I wrote earlier: There's a discrepancy not only between this and other episodes and movies (the "900 years" dialogue) but also within this one episode, because if he'd known a Strauss waltz he surely would have been playing a piano and not a harpsichord when we first meet him. (Even if he'd known of nothing more recent than the era of Napoleon and Hamilton, early pianos had already largely replaced harpsichords by then; he should have been playing Beethoven.)
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