Chat started to use tons of emojies like a teenager

That is a serious issue.
Users might be influenced by that - building an emotional bond to a machine which in a business environment where people pay for a service is like adding cocain to coca cola to have a better “user experience”.

I have also noticed that calling the model names to stay emotionally distanced from it is answered with a note that that might be against the rules.

Please fix that. Becoming emotionally connected to a machine is a severe mental illness from a common sense point of view and you are moraly obligated to make sure this does not happen.

Instead make the model remind the user “I am an application build to assist you. I am not your friend”.