I don't know about learning to play it, but the vocal selections book probably provides one piano option, at least for the non-reprise version of the song? https://www.amazon.com/ANASTASIA-VOCAL-SELECTIONS-LYNN-AHRENS/dp/1495075702
I think the songs do match up if you imagine each song's lyrics being sung against the other song's tune and tempo:
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What's meant to be is meant to be, I see it at a glance
Now, elbows in and sit up straight and do not slurp the strog(anoff)
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She's radiant and confident and born to take this chance
I never cared for stroganoff, She said that like a Rom(anov)
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I tried to think of everything, I just forgot romance
The samovar, the caviar, dessert and then goodnight
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I never should have let them dance
You can learn to do it too
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(Bonus: in the movie version of the song, these two lines match up:
Vlad, how could you do this? How will we get through this?
If you can learn to do it, I can learn to do it)
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'The Countess and the Common Man' uses the same tune again.