It is possible to duplicate any kind of projects on your BIMcloud Manager, however, it might be unnecessary.
Migration process of Teamwork projects usually goes one of these ways:
1. Re-share project on new Server
You ask all of your colleagues to Send & Receive all changes, then save a PLN from the latest version of the project. You open this PLN in ARCHICAD 23, which will do the migration, then you can re-share it to BIMcloud Basic for ARCHICAD 23.
2. Teamwork migration process
The other way is to ask again all of your colleagues to send in all changes. Then you go in ARCHICAD 20 to Teamwork > Project > Migrate Project > Prepare for Migration... When you've done all the steps the wizard asks, you will see that the project is locked on the Manager web interface and users cannot work or join. Now you can switch to ARCHICAD 23, and choose the Teamwork > Project > Migrate Project > Migrate Prepared Projects... option. This will show all possible project to migrate to the new BIMcloud Basic for ARCHICAD 23. The project's 20 version will stay locked, but you can easily unlock it when the process is finished.
None of the data will be deleted in any of the above processes, though it might be easier and faster to choose the first one.
Since you need to install a different BIMcloud Basic for 23 and 20, as it can be locked to only one ARCHICAD version, you can parallel work in both versions if you wish.
In case of BIMcloud, this process a bit easier, as you don't need to install another BIMcloud, but even there version 20 and 23 can live together without any issues.