So, according to your advice, we shouldn't use micro-services that write to a database, ever? That doesn't make any sense to me. Multiple services writing to the same database can be bad, but a single service storing persistent state in a database is perfectly fine. Just because we're micro- doesn't mean we have to cut out 90% of what a service is.
Just like all the other authenticated APIs in the world: you get a token when you log on, and use that token to authenticate yourself on future calls to the services. That's a lot of what OAuth and its ilk handle.
This management of API boundaries is likely handled for you by an app, though, so from a user perspective the story is still "open netflix, enter password, watch movie".
But how is the data shared? E.g. when you sign up you store your data. But to edit it, or to just display it you need to access it again, but that data only belongs to the sign-up microservice...?