I want to take portraits in lighting that will easily be in "range" for both cameras. If we assume ISO 200 and use lenses with similar focal length and aperture (say Nikon 50mm f/1.8 and Fuji 56mm f/1.2 set to f/1.8), can I expect the images to essentially be the same quality? Or would there be some significant difference that I am not currently seeing?
Essentially you are talking about years of technological evolution. Sensor technology has improved a lot between the Nikon D90 and Fuji X-E3 and you will certainly see it.
Expect at the very least the same per-pixel quality yet with twice the resolution. The X-E3 will certainly have more dynamic-range and color-depth too, assuming it is the same sensor or better than on the X-Pro2, which I have reviewed on Neocamera. You can take a look at the Sample Images page to see what this sensor is capable of doing. It will be a few more weeks before I see the X-E3 though.
Not only is the resolution much higher, the X-E3 uses a unique-to-Fuji X-Trans CMOS III sensor that needs no anti-alias filter without being prone to moire. On the other hand, the D90 has an anti-alias filter that softens light reaching it to avoid moire.
The lens will make a huge difference too. The Fujinon 56mm F/1.2 is extremely sharp, more so than the Nikon 50mm F/1.8G, even wide open. It also has almost no distortion, although there is slightly more vignetting at F/1.8-F/2.2 than on the Nikon. Even the Nikkor 50mm F/1.4G is not as sharp as the Fujinon 56mm F/1.2.
While you asked about image quality, you should know that these cameras use completely different autofocus systems. On-sensor AF of the X-E3 is extremely accurate but based on my review of the X-Pro2, it becomes noticeably slower in low-light.
For the example you give, by far the biggest difference would be from the lenses. The 50mm f/1.8 is Nikon's cheapest lens and you're intending to use it wide-open - which means you're using an lens with mediocre image quality1 at the point where its image quality is lowest. Compare that to the $1000 or so Fuji 56mm which you're using down a stop and a bit from its maximum aperture. Shockingly enough, a $1000 lens is better than a $100 lens.
1. Mediocre in terms of interchangeable lenses. I love my (Canon) 50mm f/1.8 and it gives me images that a compact camera could never give me, but there are lenses with better image quality in almost every way out there. Just not for $100.