How do I transfer a signal install (verification and all) to another computer (both are Linux)

I don't believe signal let's you import any of your old messages to new device's for security. When setting up a new device last time I saw a warning about this. It let's you use the same account but without the older messages.

You could try to copy your ~/.config/Signal folder to the same location on your new device but that might not work. Don't know because I've never tried.

If I remember correctly just scan the QR code.

I use the flatpak version, which keeps all of the cache and config directories together, and I'm fairly certain I've simply copied the flatpak data folder between computers in the past. I could be wrong though; I might have only copied it between distros on the same hardware.

Pure guess here...

signal uses "magic" to encrypt messages at rest.

Part of this magic relates to the hardware.

Change the hardware and you won't be able to decrypt the messages.

So try to move a message database to a new machine (either because you want to or because someone else has snaffled it) and you won't be able to decrypt the database.

You can't transfer the desktop App at all. But you can transfer your phone easy enough. Just go to account, scroll down and choose transfer account