"Interpolate" is a more genteel term than "copy" or "mimic", I don't think there's anything more to it than that. Maybe it's a little less legally triggering, but in the end they mean the same thing.
But I've never come across any definition of "Interpolation" that actually describes it that way, or that could (in my opinion) reasonably be interpreted as describing the modern use of the word.
I'm sure future editions of established dictionaries will include the modern use amongst its definitions, because it's in popular use (even if it turns out to be a misuse), and it's by now been used in the modern context in print media (both of those things being criteria for inclusion in the Oxford English Dictionary).
I was just curious when the first use / misuse was, and if it had appeared in the Gilbert O'Sullivan vs Biz Markie case (whether Markie's legal team tried to claim a defence of interpolation, or O'Sullivan legal team eliminated interpolation as part of their case), which would make sense both in terms of people citing that case, and interpolation having a legal definition that covers the modern usage.
It's not just for sampling any more. A lot of the oldies music played over the speakers in big-box stores and chain restaurants these days are interpolated performances. For background music it's cheaper to hire people to exactly mimic a hit single than to pay the fees for the original recording. Which says as much about how little musicians are paid these days as it does about how much record companies are demanding.
This is nothing new : throughout the late 60's to the mid-80's, rather than buying full priced hit singles, you could instead buy "Top of The Pops" albums every couple of months, for a fraction of the price, that comprised pretty faithful recreations of current top 40 hits.
Bizarrely, not only were some of the tracks performed by future hit makers (such as Elton John) before they were famous, but in a truly meta moment of pop will eat itself, some of the records were so popular that they charted in the top 40 (where I assume a smaller, cheaper label decided to cash-in by selling albums of cover versions of albums of cover versions...)