Double-clicking the pause/play button on the headphones to skip a song does not work. All other functions pause, play, volume work. I'm using the USB-C headphones that came with the Pixel 3. Skip works in other applications like Google Music. I can reproduce the problem with Samsung headphones + USB-C adapter.
Several of you have reported back saying that setting up Google Assistant resolves the issue. As a result, we'll be closing this thread out.
If you're having trouble setting up Goggle Assistant, please refer to @CJSelf36's reply below, which walks you through the steps.
Apologies for the delay! Been distracted with festivities...
To answer @Peter's question: Yes setting up Google Assistant did resolve my issue.
When turning on your Pixel 3 for the first time, you'll be asked to setup your Google Assistant during the "Welcoming stage" (as I like to call it..). However, if you're special like me and chose to skip this step, then you will have to go back later on to properly turn on your Google Assistant. The easiest way to achieve this is by downloading the "Google Assistant" app from the playstore.
If you have Google Assistant setup (and working), and the skip function is still failing (@JanisOzolins), then I suggest voice commanding your Google Assistant to skip the song for you (by holding middle button while saying "Please skip the song my dear Google" [Not sure if already mentioned, but AI is much more compliant with kind manners..])
After passing the command and successful, try double-clicking to test the skip function. If your command was unsuccessful (and Spotify is currently playing a song on one of your playlists), then your Google Assistant is dysfunctional. In that case, giving feedback to Google is your best choice (Settings/System/About phone/Send feedback about this device).
IF...your Google Assistant is properly setup, and you were able to pass the skip voice command, BUT the skip button is still not working then I suggest the following:
Make sure your system is up-to-date
Make sure your Spotify app is up-to-date (maybe even try reinstalling the app)
Definitely reboot your phone
If no luck after that...then it is truly up to the gods.
As the Late Great Marcus Aurelius once noted, “When the force of circumstances causes you, in some sense, to lose your equilibrium, return to yourself with all speed, and never lose the rhythm for any longer than you must; for you will be more in control of the measure if you return to it again and again.”