Which metrics should a product owner look at?
I cannot list everything you need to monitor for a database. The biggest factor is 'It Depends'. I am going to list a few articles that might give you a starting point and from there you can adjust.


SQL Server Monitoring Tips--I know you are not looking for how to monitor but what to monitor. If you go thru these categories and cover all you will cover most of the KPI.
Key metrics for SQL Server monitoring by Paul Gottschling
Server Performance and Activity Monitoring