I know it can happen with upgrade, but I have reasons to install a clean 2x and then to transfer the database.
Is there a converter, like (example) IPB to SMF?
No, there isn't one called converter. But the upgrade pack is EXACTLY the install pack (so as fresh files as you can get :)), only it doesn't have the Settings files (so it doesn't overwrite yours) and it has a few extra files, the upgrade.php and upgrade sql files. That's all, for the rest is exactly clean files - those from the installation pack.
In addition, the only way to get the database itself from 1.1.x to 2.x is to upgrade it with this aforementioned upgrade.php and its sister sql files. (well excluding manually editing the database :))