I worked in a school portrait lab for 33 years, 1979 to 2012. I saw everything from full film and optical printing to scanned film and digital printing to digital capture and digital printing. In the film/optical days, we used Kodak Vericolor III and later Portra 160NC films, and printed on Kodak Portra Endura paper. That combination evolved, but it had a consistently neutral look and a rather low contrast appearance, compared with Kodacolor or Kodak Gold. With Kodak Gold and Royal paper (typically used in mini-lab operations), you got lots of color saturation and higher than normal contrast. Kodak always thought that was what the general public wanted to see, at least for general snapshot work.
Our goal was always accuracy. When we started scanning film, we used nine high speed, high resolution Kodak Bremson scanners. I set them up, calibrated them, created custom film terms for them, and generally matched them to the look of a film negative printed on optical printers to Portra Endura paper.
When we started using digital capture, we tuned our cameras’ JPEG output to match that same look as closely as possible. After a season working with a couple million images, we fine-tuned that look to taste, and stuck with that look until the lab was sold.
Yes, there is a Fujicolor look, a Kodak Gold look, a Portra look, etc. It is unique to the particular brand, type, speed, and even the emulsion batch of film, although the manufacturers do stay within a reasonable tolerance range for spectral response of a given film. If you get the neutrals right from shadows through highlights, you will see the unique spectral response of a film. Some films respond differently to different light sources. Some are heavier in the greens, while others are heavier in the flesh tone parts of the spectrum. But if you calibrate for a full tonal scale and get it neutral from black to white, the film will scan the way it is supposed to look.
In the end, though, you can get any color balance you like. Just tweak the sliders to taste. But first be sure your monitor is correctly calibrated and profiled, and that you use a simulation profile or proofing profile when making final adjustments.