Not sure of the model as I have had it for over 10 years.
There is just one model that fits that description: Apple TV (2nd generation) “A1378”, which was sold from Sept. 2010 to March 2012. The model before that didn’t support AirPlay. The model after is at most 9.6 years old.
Once I contect and hit the TV logo in lower left cornor,
The AirPlay symbol may be in the controller bar at the bottom of the video, either left or right. Or not at all. Depending on the video hosting service they used.

It acts like it is connecting to the Apple TV and the I get a error message that it failed to connect and try again.
Verify that your devices are all on the same network. If your iPhone or iPad is not Wi-Fi connected, then it may not find the Apple TV. Try restarting your devices, including the router. For the Apple TV: Settings﹥General﹥Restart.