Yes! That has been supported for a long while. At least on Android, go to Settings -> Chats -> Chat Backups. Set up a schedule and a passphrase and a folder, and it will export your chats every day.
I do that and then sync that folder with another computer using SyncThing.
From the point of view of iOS, yes it can (the person you're replying to is wrong, as explained by the other person who replied to them). But no, the Signal iOS app does not currently have that functionality.