Is there a service that could track down a melody that has been duplicated by someone else? [Edit: Not Shazam, that app never finds anything for me]
Could that other person be sued for using a remarkably similar melody?
Yes.
The melody is the most “copyrightable” part of music there is. Now get back to writing songs. 
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Yes and yes. Melodies are the most important part of the identity of a piece of music. It’s what’s considered first and foremost in copyright law.
Conversely, one cannot copyright a chord progression or ‘style’ of music. The only way a chord progression could be arguably copyrighted is if it’s completely unique and serves the role of a melody.
No, I doubt there’s any software on the planet capable of detecting a melody accurately.
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Thanks for the replies, you’ve both solved the question
I now realise the obvious: Enseek / AdRev etc. rely on some sort of fingerprinting done in-house by themselves, and seek out that fingerprint in uploads to their designated corner of the internet (VideoHive, YouTube etc.). They don’t do a melody search as such.