Is there an iPhone-like Keychain in Android?

You you use storage on the SD card like Michael Cramer stated ( Let the user know you are storing data :-) ). The data on the SD card is not deleted when the app is removed.

You could on the other hand use a server which stores all the details which can be downloaded as and when required and instead of using a user name use the IMEI of the phone which is unique. That way even if the user reinstalls the OS he will still be able to refetch the settings.

for getting the IMEI you may refer : How to get the device's IMEI/ESN programmatically in android?

You have the real thing here:

By integrating Smart Lock for Passwords into your Android app, you can automatically sign users in to your app using the credentials they have saved. Users can save both username-password credentials and federated identity provider credentials.

Integrate Smart Lock for Passwords into your app by using the Credentials API to retrieve saved credentials on sign-in. Use successfully retrieved credentials to sign the user in, or use the Credentials API to rapidly on-board new users by partially completing your app's sign in or sign up form. Prompt users after sign-in or sign-up to store their credentials for future automatic authentication.

https://developers.google.com/identity/smartlock-passwords/android/