There is probably some way to determine if the request was denied automatically by the browser or manually by the user (e.g., time to get "response"), which is definitely something which can be used for fingerprinting.
Which reminds me of fingerprinting by tiny differences in the audio API provided by browsers [0]. Super interesting, but also a bit depressing. Also works for things like canvases and WebGL.
EFF allows you to check how fingerprintable your browser is [1]. Do note that the results may not be very accurate.