You are asking how it might be done so the following narrative is somewhat speculative but sound.
You don't need the original tapes but the process would be quite involved. We have (or can make) frequency-time domain profiles of the instruments and use match filters to find them in the mix. Subtract them out and repeat.
The model signals could be from a library or you can make them straight from the recording itself if there are segments where the instrument(s) are playing solo. Actually the complement might be an easier process. Focus on the one thing you want, filter it and throw everything else out. The idea is not just to get the notes played by the instrument (if you have a score you can use that too), but to get the acoustic fingerprint of the instrument. One advantage to using the recording to sample different instruments is that you get the effects of the room and everything else included.
This process may need to be done multiple times on a single instrument and errors can be introduced. Some type of adaptive filtering could be employed where the filter parameters are refined with each try. In some cases you don't need a sophisticated model or a perfect sample (which you cannot get anyway).
It's not easy but it is possible. Also, if there was only one instrument and a voice and there are blocks of time where they don't overlap this makes it easier.
It is not likely possible to get a perfectly clean voice or other instrument pulled from a recording. There are other uses of this like narrowing in on a conversation recorded in a crowded place. The separated voice will have some remnants of the other voices present and other sources of noise. But if it's clear enough be coherent various noise reduction techniques can be used to smooth out the voice track. Since it will be used as an overlay to an orchestral recording the left over noise will not be too bad.
As Tim pointed out, if you have isolated tracks this is a very easy operation.