Hi @Tunkkari ,
Thank you for your post.
Could we just restore the current Reporting Services 2014 databases to the new SQL Server 2019 -server
The report server database contains published reports, models, shared data sources, schedules, resources, subscriptions, and folders. It also contains system and item properties, and permissions for accessing report server content. If your migration includes using a different Database Engine instance, you must move the report server database to the new Database Engine instance. For more information, please refer to Move the Report Server Database.
install the SQL Server Reporting Services 2014 to the new Reporting services -server, make the necessary configurations to Reporting Services and continue using the Reporting Services 2014?
I made a test as below:
1.Install SQL Server Reporting Services 2014 with SQL Server 2019

2.Change the data source to SQL Server 2019

3.Change the TargetServerVersion to SQL Server 2008 R2, 2012 or 2014, and then deploy project. It's succeeded.

In addition, some SQL Server components or services are specific to an instance of SQL Server. These are also known as instance-aware. They share the same version as the instance that hosts them, and are used exclusively for that instance. Examples: Database Engine, Analysis Services, and Reporting Services. For more information, please refer to Work with multiple versions and instances of SQL Server.
Best regards,
Carrin
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