I tested whether it was consistent between iPad, desktop, and my Yamaha amp which acts as a smart speaker. It definitely did not carry the setting over, and Spotify confirmed that behaviour.
I just tested using Spotify on my iPad and my iMac, listening on an Echo Dot, Echo Show, Denon AVR-1913, and Google Home Mini.
The Echos use Spotify Connect, the Home Mini uses Google Cast, and the Denon uses Spotify Connect from the iMac.
From the iPad the Denon was not showing up in the Spotify devices list so I used the option to connect to an Airplay or Bluetooth device to connect via Airplay. Later, I noticed Spotify was asking for permission to scan my network for devices, I gave permission, and the Denon then started showing up directly in the devices list. I don't know if connecting to the Denon from the device list uses Spotify Connect or Airplay, because mobile Spotify unlike desktop Sotify doesn't say on the device list how it connects.
The test was to tell connect to the test target, start playing a one song playlist, then when it starts playing use the position slider to skip to near the end, switch to the view that shows what is playing and the queue and up next lists, and see what happens when the song ended.
Then I'd stop it (if it was playing), disconnect from the target, toggle the autoplay on remote devices setting, and do the above test again.
I'd again stop it (if necessary), disconnect, and toggle the autoplay on remote devices setting back to its normal setting (which for me is off).
For the Echo Show, Echo Dot, and Google Home Mini it obeyed the autoplay setting. When off it stopped after the playlist ended. When on it started playing a list of things it felt were related to the playlist song.
For the Denon it always stopped when the playlist finished, both when I was definitely connected with Airplay and when possibly connected with Spotify Connect.
Make sure you have an up to date version of Spotify. They added the ability to control autoplay on remote devices 2 or 3 months ago.
The Echos use Spotify Connect, the Home Mini uses Google Cast, and the Denon uses Spotify Connect from the iMac.
From the iPad the Denon was not showing up in the Spotify devices list so I used the option to connect to an Airplay or Bluetooth device to connect via Airplay. Later, I noticed Spotify was asking for permission to scan my network for devices, I gave permission, and the Denon then started showing up directly in the devices list. I don't know if connecting to the Denon from the device list uses Spotify Connect or Airplay, because mobile Spotify unlike desktop Sotify doesn't say on the device list how it connects.
The test was to tell connect to the test target, start playing a one song playlist, then when it starts playing use the position slider to skip to near the end, switch to the view that shows what is playing and the queue and up next lists, and see what happens when the song ended.
Then I'd stop it (if it was playing), disconnect from the target, toggle the autoplay on remote devices setting, and do the above test again.
I'd again stop it (if necessary), disconnect, and toggle the autoplay on remote devices setting back to its normal setting (which for me is off).
For the Echo Show, Echo Dot, and Google Home Mini it obeyed the autoplay setting. When off it stopped after the playlist ended. When on it started playing a list of things it felt were related to the playlist song.
For the Denon it always stopped when the playlist finished, both when I was definitely connected with Airplay and when possibly connected with Spotify Connect.
Make sure you have an up to date version of Spotify. They added the ability to control autoplay on remote devices 2 or 3 months ago.