Failover clustering SQL server 2016 standard
Please have a look at Features Supported by Various Editions of SQL Server.


  Or put differently (perhaps?) is failover clustering always on the entire instance and all databases it contains?


No, AlwaysON Availability(AOAG) groups can be configured for specific set of databases. It is not necessary to include all databases which reside on instance. So suppose you have 7 databases on instance you can configure AOAG for 4 databases and leave aside 3 databases.


The unit of failover would be the AG group, this means when failover will happen all databases which are part of that AG will failover
One or multiple databases can be grouped within an Availability Group
One SQL Server instance can support multiple AGs. The Availability Group is the unit of failover
A specific database can be part of one Availability Group only and cannot be a part in multiple AGs


Please read more about AlwaysON Availability groups from this blog it has many links start with link "What is AlwaysON"

Basic AG would only support one database.