The core problem here is the hotel Wi-Fi. As I mentioned, hotel Wi-Fi networks often require you to login to be able to access the internet, known as captive portals. Chromecast, and TVs with Chromecast built-in do not support those Wi-Fi networks, hence your difficulty adding it to your Google Home app.
To add a TV with Chromecast built-in to your Google Home app requires your phone and the TV to be connected to the same Wi-Fi network (i.e. the hotel Wi-Fi). However, even in the case you did that, those kind of Wi-Fi networks nearly always have a security feature enabled called client isolation, which prevents different devices from being able to communicate with each other (i.e., your phone communicating with the TV. Hopefully, that explains why fundamentally Google just doesn't support Wi-Fi networks that require you to login, known as captive portals.
With all that said, there is a workaround that users have found success with, which is you can can try to use a phone or tablet as a Wifi hotspot. However, this will then use your mobile data plan, which could be expensive for you depending on the data plan you have with your carrier. Note you would need 2 devices to set this up.
- Turn on the Wifi hotspot feature on a phone or tablet. If you're not sure how to do this, refer to the manufacturer's support website.
- Connect a second phone or tablet to that Wifi hotspot.
- You can then use that second phone or tablet to setup the TV and connect it to the Wifi hotspot on the first phone or tablet.
In this way, both devices are connected to the Wifi hotspot under your control, thus no issues with logging in for internet access, nor security restrictions that prevent devices communicating with each other.
