According to Wikipedia, there was a 1956 movie based on a 1952 French play based on the idea that Anastasia survived. Also, the phenomenon of people claiming/pretending to be Anastasia is real, and the play is based on the most famous of these. (The story continues to evolve: a 2016 stage musical included some of the same songs and characters, but substituted or added others.)
The rags-to-riches plot of this 1997 movie makes it feel like a standard Cinderella tale, but now that I think about it, the story may be more like Pygmalion (or My Fair Lady): a man fabricates an identity for a woman, then falls in love with her. That arc fits better with the 50s versions, though: in the 90s cartoon, the woman really was a princess all along. (Edit: It’s not my imagination… again, according to Wikipedia, they actually considered adapting My Fair Lady and deliberately took elements of Pygmalion. But I noticed the similarity myself.)
