I use the convention that all my containers are getting a /data/ folder where data can be stored and will get an automatic daily backup.
It is easy to set the right storage behind /data/.
Just a convention, nothing more, smoothly working for the past "many" years.
I assume you're saying this as a container image author, not as someone who is deploying containers. It'd be great if every other container image author standardized on something like this. We just need someone to create the standard and some tools to make the standard a well-paved path.
Some sort of "hey back this data up, it's important to my function standard would be nice, and not just for container apps.
Trying to figure out where your app hid its configuration and save files is a perennial user frustration.