Added March 2019: the "Account Key Backup" seems to have been removed from Messenger, so I have updated this answer.
According to a Messenger help page,
In order to switch to a different account or access your account on another device, you'll need to add a password. To add a password to your account:
- From Chats, tap your profile picture in the top left corner.
- Scroll down and tap Password.
- Tap to enter a password and tap OK.
You should now have a password associated with your account and can Switch Account
The Messenger help page currently states "This feature is not currently available for Android" but it is present in the Android app.
Old solution: (cannot be used anymore)
To transfer Facebook Messenger conversations to a new phone without a Facebook account, while keeping the same phone number, do the following.
On the old phone, open Messenger and tap your profile picture to go to your profile settings.
Tap "Account Key Backup".
Select the option to back up your account key to Google Drive. After the backup has been made, Messenger will say
Account Key Backed Up - Your account key is saved to Google Drive with the email address ***. You'll be able to sign into Messenger if you switch phones or uninstall the app.
On the new phone, install Messenger, and open it.
Select the option "I do not have a Facebook account", and enter your phone number. You will receive an SMS with a verification code to enter in Messenger.
Messenger will ask you if you want to continue with your existing account; answer yes. Messenger will tell you that an account key has been found on Google Drive and use it to copy your conversations. However, secret conversations won't be copied.
You will be logged out of Messenger on the old phone, and will no longer be able to use Messenger there.