I have a Nokia Lumia 625 and it is updated to Windows Phone 8.1. In settings it also has a "Project my Screen" option.
If I buy a Chromecast 2 device, will I be able to cast videos or the phone's screen to a TV?
The "Project my screen" feature in Windows Phone uses the Miracast standard. It's not compatible with Chromecast, which uses a different technology altogether.
If you want to stream media from your phone to your TV, you should get a Miracast adapter. I'd recommend the Microsoft Wireless Display Adapter, but there are other options as well.
For more information, see How to use Project my screen in Windows Phone 8.1?