You can make a chat backup, but it's an encrypted binary blob, not a mailbox file or a JSON file or some other more accessible format.
I think I understand why they do that (if you send someone a message on Signal, they try very hard to make it difficult for anyone but the recipient to read it, whether that's by intercepting traffic or reading data stored on your device, or rummaging through your backups) but it does make it a bit of a pain.
Seems short-sighted. The more people who use Signal, the more secure their users' chats will be because they can use other apps less. Omitting basic features for "security reasons" reduces security overall, because you can't force other people to use Signal.