I know in 1993, Adobe software was not available.
How was this effect accomplished?
As I mentioned in the comments, movie editing software certainly was available in 1993 (see Jurassic Park for example). That being said however, I was under the impression that Schindler's List was deliberately filmed by Spielberg on black and white film, in part to prevent a possible future release of the movie in colour (which would have gone against his vision of the story portrayed by the movie).
Apparently this is correct - the movie was shot on black and white film by Janusz Kaminski - a cinematographer who has shot all of Spielberg's films since 1993. However, according to this article:
Kaminski shot most of the film on black-and-white emulsion, save for the sequences featuring the little girl with the red dress, which were shot in color emulsion and then painstakingly desaturated in a process called rotoscoping, which Kaminski describes as "an old version of CGI, except each frame was done by hand."
Rotoscoping is a technique where the area to be preserved is masked off, and the remainder of the image is worked on in some fashion, frame by frame. The technique has been around long before 1993 - it was used to great effect in the music video for A-ha's Take On Me, released in 1985, to pick but one example.
It is called Recoloring.
In the color correction you can "highlight" a color and set everything else black and white (saturation to 0). This is technique for very long time already.