I don't have an actual answer for you, except to say that both the remote and the chromecast work with bluetooth, so it'll probably connect and give you basic functionality.

I don't know if advanced features will work. For example, will the microphone button trigger the assistant? Will the app specific buttons work?

I'm assuming, for the sake of this conversation, that you just happed to have the PS5 remote on hand. If not, you can buy a second chromecast for just ten extra dollars over the cost of the PS5 remote. You can also buy a replacement shield remote for the same price as the PS5 remote, and I'm guessing that'll work even better.

You can buy the new triangular remote direct from Nvidia, it’s a big improvement over the old one and does most of what you are looking for.