I had a very disappointing support experience with SpotifyCares on Twitter trying to troubleshoot this track-skipping Chromecast issue for days. I'm not alone in it: if you google for "spotify skipping to next song chromecast", you'll see multiple threads, some going back to 2016.
I know it's not my WiFi, or my devices (iPhone, Chromecast Audio, Chromecast video 3rd gen) as NO other apps casting to Chromecast (YouTube Music, etc.) have this problem. I had a friend over to play FLAC files off his device on my same network and Chromecast device without issue.
It's just Spotify. There's an unacknowledged bug on their end. My guess is that it's a buffer issue in the networking code, Spotify not being able to send data smoothly enough to the Chromecast, or Spotify losing contact with the Chromecast device. I've gotten the issue on the Spotify web player on Chrome on Windows (though not as often / easily reproducible) without casting, though, so maybe not. Perhaps it's just most easily seen when chromecasting.
Things I've tried:
Use Spotify's test account (to rule out issues with my Spotify account being corrupt)
Reboot WiFi router, reboot cable modem, make sure Internet connection is good (Comcast 130 mb/s down, 6 mb/s up, 12 ms ping)
Check for firmware updates on the Chromecast, OS updates on iPhone, Spotify updates
Uninstall, reinstall Spotify app
Change WiFi from 2.4 GHz to 5 GHz
Try songs in and outside of playlists, try downloaded songs versus streaming
Reboot AV receiver, reboot phone
Try Chromecast audio device instead of Chromecast video and vice versa
That's just what I can remember. There's probably other things I've tried. This isn't one of those "gee, have you tried X obvious thing?" Yeah, I have.
I'm cancelling my Spotify Premium subscription today. I wanted to put this up so people googling for this issue have more data and know it's not just them.