Can I use the Spectrum TV app on multiple Apple TV's on different Wi-Fi networks?

As long as your WiFi networks are on the same Spectrum internet service, yes. If they are not on the Spectrum service network you can still use the Spectrum app but you will be limited in what you can view. Spectrum explained to me that it’s up to the channel provider as to whether they will allow viewing outside the Spectrum network. I have two Apple TVs in the home, both running the Spectrum app without issue, but they are on the same WiFi network.

Looking at Spectrum’s website, yes, it appears you need to be connected to your own Spectrum Internet Connection.

https://www.spectrum.net/support/tv/get-started-spectrum-tv-apple-tv

Unless Spectrum somehow prevents that, you should be able to yes.

Spectrum TV is not an Apple product, I would recommend that you contact Spectrum to ask about their products.