It really depends on the cameras.
It always depends on the cameras because the sensor's pixel pitch is an attribute of the camera. For two cameras with the same number of pixels, the magnification ratio matches the crop factor because the ratio of the pixel pitch is the same as the ratio of the crop factor.
For two cameras with the same pixel pitch, the magnification is equal--the size of the sensor doesn't matter. In fact, if the sensor tech is identical and the same lens is used, one camera may capture more of the scene, but the portion captured by both will be identical in IQ, detail and whatever else you want to measure.
I much prefer a high res full frame camera for wildlife (mainly birds). I find the cropping options, iq and detail better.
Full-frame cameras generally get the best tech, so they are likely to have better IQ. How would a crop sensor with a smaller pixel pitch but equal tech compare? Hard to say--you're trading spatial resolution for light gathering. In good light, the smaller pixels might do just as well and provide greater magnification.
Generally, people aren't comparing apples to apples. Even related cameras from the same manufacturer may have differences beyond pixel pitch and sensor size that make real comparisons impossible. And I find that the people who have FF cameras for birding also seem to have expensive lenses to go with them.
As I like to maximize magnification, my ideal camera would have a pixel pitch as small as I could get away with given my lens's ability to resolve detail. Whether this ideal camera would be FF or not would depend on whether I could afford a FF version and had the space for the larger files.
One place FF is a clear winner for bird photos is birds in flight. For any given lens, a FF has the biggest FOV, which makes it easier to keep a flying bird in view.
It's a good thing we have choices!