What kind of messages are those used by Bitcore When creating a transaction proposal with Bitcore,
the options include a "message" field, both for the transaction and the single outputs.

Are these messages those used by the Bitcoin network peers (like 'ver', 'inv', etc..)?

Where are these messages stored?
This is the answer I received from a Bitcore developer (see here)

Messages are stored in the BWS server database encrypted by the wallet encryption key.
Therefore, the BWS server maintainer can not see any of the messages, but the copayers can see it because they have the decryption keys.
These messages only exist on the BWS server, so if the BWS server disappears. So does all your messages.
Wallets can be restored outside of BWS, but messages are tied to the BWS server.