I'm not sure if this is a feature I got grandfathered into, but I've got WhatsApp set to back my messages up to Google Drive every week. On my Android phone, the setting is in settings -> chats -> chat backup.
It's something they still push aggressively - on Android. For iOS, I believe they use iCloud, and only iCloud; this means you can switch between Android handsets, or Apple handsets, but not Android to Apple (or vice-versa).
Note that this is based off what I've seen and heard from others, however - I've never actually owned an iPhone to test this.
Note that this is based off what I've seen and heard from others, however - I've never actually owned an iPhone to test this.