Can a smartphone light meter app be used for 120 format cameras?

Speaking strictly about setting an exposure (leaving out depth of field):

Pre digital, when 35mm, medium format and large format film was in normal use, you were often using a hand-held physical light meter - similar in function to your app I would imagine. I've never seen one that had a setting for the size of film you are using (I've been taking pictures since the 1980s, including medium format and large format photography, and with both antique and "modern" hand-held light meters).

The f-stop system should give us a set of numbers that don't need adjustment for film size (I'd almost say it was designed for that, but I can't find a reference). The best way to think of the f-stop is as a theoretical transmission of light on a unit area basis. So not the total amount of light sent to the film.

Note "theoretical" - real world lenses don't send 100% of the light hitting the front of the lens to the film (or sensor). The amount lost depends on the lens coatings, complexity of the internal design and build quality. But the loss is usually not that important for film still photography.

Your 35mm Nikon and Fujifilm camera both have on-board light meters that look through the lens and automatically compensate for the loss of light through the lens (in addition to perhaps doing other things, like compensating for light or dark subjects). The newer Fujifilm may be doing more compensating than the Nikon is capable of, which could make it better or worse as a light meter.

My advice would be to shoot a roll or two where you shoot three frames for each subject, one metered by the Fujifilmm, one by the Nikon and one by the app. Keep good notes about the settings for each frame. When you get the results back, you may see which metering technique works best -- or maybe even which works best for for specific types of shots. Make sure that both cameras are set to manual ISO and are using an ISO that matches that of your film and that the light doesn't change much between exposures.

Light meters don't care about camera formats, they just measure light. Here's a Gossen Lunasix 3 . I still have one. There's no setting for format.

Keep in mind how you are using the light meter. There are two modes, Incident and Reflected. Built in camera meters are Reflected meters, they implicitly assume a certain level of reflectance, (I'm trying to avoid the whole 18% war). This of course doesn't work accurately if your subject differs greatly in expected reflectance, "snow" for example. Using a meter in Incident mode, measures the amount of light falling on the subject. There is no assumption about subject reflectance.

A Through-The-Lens (TTL) camera meter is also looking through any filters you may have on your camera, not so the light meter.

Reflected light meters, both in camera and standalone, have an angle of view, sometimes several selectable views. The computed light level over different views may differ significantly. In fact the Fujifilm X100T description says there are three metering system on the X100T which is multi, spot, and average. Depending upon the subject, lighting, and mode, the readings could easily differ a great deal.

Caveats: Most phone cameras are moderate wide angle lenses. Your app may have a feature to use the light within a certain part of the phone image.

Historically there were "Averaging meters" which took an average of the whole scene. You got in big trouble if part of the scene included the sun. Then there were "spot meters" which had a narrow field of view, and looked like an early phaser. You'd aim at the face of your subject, or you'd hold your hand up so that it was lit the same way your subject was, and take a reading off your hand.

These were both reflected light meters.

An incident meter had a cute white hemisphere over the sensor and you put it where the subject was facing the camera. You were measure the actual lighting that hit the subject. Movie makers preferred this, as it made the shots more consistent.

TL;DR: Try it. Should work. If your phone app is an averaging type, then take your reading much closer to the subject if the lighting is contrasty. In effect, fill the frame of the phone roughly the same image as you will use on the camera.

Simply, YES, light readings are universal.