Do a class equivalent to IOS's SKStoreReviewController exist for Android?

I'm afraid Google has not yet provided any such official API's. There's no way one can submit reviews/ratings without leaving apk.

Android provides now an API to make a review directly from app.

"The Google Play In-App Review API lets you prompt users to submit Play Store ratings and reviews without the inconvenience of leaving your app or game. ... to protect user privacy and avoid API misuse, there are strict guidelines that your app should follow about when to request in-app reviews and the design of the review prompt."

more information > https://developer.android.com/guide/playcore/in-app-review