How did you keep your heart?
I think here it may mean "how did you keep your determination or strength". It sounds like she went through some hard times, but still was able to maintain the strength (heart) to keep going.
I personally wouldn't say it like they did in the movie. It might be more relevant to a certain era. (the background maybe looks "old"?)
This is what I would have guessed if I didn't just see the movie.
Lily, in this context Natasha has "kept her heart" in that she is still nostalgic, sentimental, appreciates the good times with her fake family, even though it wasn't real. She was also always hopeful she would find her real mother. Melina went though the red room so many times and was dehumanized to the point of being all about the training and the mission. It's all her "brain," whereas Natasha was also kept her "heart" meaning she still has feelings. The brain thinks and the heart feels.
To your other question, if one has "heart" it usually means they determination, courage, a will to keep going like Eric staid. In the context of this movie, it is being used in a very specific sense and it is trying to be extra dramatic. Also, remember these are Russian spies who learned English later. So, in a nutshell, no, you would not ordinarily talk this this, although it does make sense.
Thank you so much for the help!!!