Upgrade SQL Server, not user databases
No. As soon as you attach a database to SQL Server, assuming it is compatible, the internal system tables will be upgraded.

While this process is a one-way operation, it takes a few seconds at most.

If you do not wish to upgrade your user databases, don't upgrade to the latest SQL Server.

If you want to keep your databases on a previous compatibility level, that can be done from T-SQL code or inside SQL Server Management Studio, but for all intents and purposes, the internal version of the database will match that of the instance it is installed on.