- Apple implemented iMessage E2EE sync across devices back in 2011. But be careful not to save your chat keys in iCloud backups (local backups are fine), unless you enable E2EE for iCloud backups as well, which is an option rolled out in 2023.
- WhatsApp appears to have rolled out a form of E2EE device sync in 2023 as well. WhatsApp Web complicates the question of how secure is the E2EE though.
To be clear, Signal now allows secure backups to the cloud. If you don't use a strong password, and much as the public won't, it's not perfect but they maximize the security. (And you can always choose to not use the backup.)
https://signal.org/blog/secure-value-recovery/
(It's also relevant to the OP.)
No, it doesn’t, at least on iOS. There is no backup option on iOS. The only way to retain information when using a new device is to have the old device close by and transfer it by running Signal on both of them. Anyone who loses their device or does not have it with them when getting and setting up a new device will lose all the messages from the older/previous device.
The app is full of tiny annoyances like this. For sending photos there are some editing/cropping tools - a fun and somewhat useful innovation, but while the crop handles work from the corners of an image they don't work properly from the sides. I was a busy evangelist for the product in its early days, now I hate using it.