I've owned a Chromecast for years but now it's become trouble! First the 'TV casting icon' went missing. I had another Tv looking icon replace it (never seen before). It wants to project the image another way (Air Play maybe?). So, I said what the heck, I'll just reset the Chromecast and start all over. No problem. So, I press the button on the Chromecast and wait for it to speed up and turn orange, but it never does. Then I look at the TV screen and it says it's attempting to pair. What the hell is going on?
Resetting can be a bit tricky. While the Chromecast is plugged into the TV, hold down the button on the side of the Chromecast. The LED should begin to blink orange. When the LED light turns white, release the button and the Chromecast should then restart.
A photo of where the button is by model:
Chromecast FDR.pdf - Google Drive
Alternatively, if you can still see it in the Google Home app:
To your original question, the TV looking icon you referred to is AirPlay, which is an Apple proprietary sharing technology, not the same as the casting icon you're used to seeing. This only works with iPhone, iPad and Mac devices. Your TV would also need to support AirPlay for you to use it.