Cipherchat uses Lightning Node Connect which is used by many bitcoin apps including Zeus, Alby etc. It is not possible to link your identity to your node with LNC. Also the whole stack can be self-hosted. All of this is explained in the FAQ and Privacy page.
Who's saying to share your pubkey with anyone you don't want to know it? Do you give your phone number to people you don't want to contact you... The fundamentals of privacy is sharing information with people you choose, that choice is up to you.
Pubkeys are also publicly available so it gives a way for nodes to communicate with eachother without knowing any other information. It can be very useful if you need to coordinate channel closures or notify peers of upcoming downtime for example.
Yes, pubkeys are public and used as identity within the LN protocol. But as soon as you start to use pubkeys in other protocols, you attach additional significance to them which can be used to deanonymize. Again, there is no technical reason to use LN pubkeys in a chat protocol.
02e64da75794703de72e9a8d6394747bc40fd72d0121c5dda6c1bd845939e30bb7if you want to try it out!