To resolve some of the mystery here she almost certainly has perfect pitch, meaning she can identify individual notes by sound without a reference pitch (like most of us require).

She's also an extremely talented pianist with oodles of experience at the instrument.

As she hears the melody she instinctively understands what chords would work underneath and is more or less 'reverse-engineering' the songs on the fly similar to how the original composers would've written it. With catchy/pop/theme songs the chord progressions themselves will be fairly predictable. If you play music a lot these things do start to jump out at you and you'll find yourself finishing the original composer's 'sentences' without needing to refer to the original work or sheet music or tabs or whatever you were learning from.

For someone with this skill set it's probably like hearing a sentence spoken out loud and being asked to repeat it. You understand the words, you know how to make the sounds with your mouth so... you just... repeat it. No big deal.

Did I mention that she's ridiculously talented and experienced? Because yeah. That.