Just digging into the React Native code, I found the answer.
Android
The React Native AsyncStoragemodule implementation is based on SQLiteOpenHelper.
The package where all the data classes are handled: https://github.com/facebook/react-native/tree/master/ReactAndroid/src/main/java/com/facebook/react/modules/storage
The class with the instructions to create the database: https://github.com/facebook/react-native/blob/master/ReactAndroid/src/main/java/com/facebook/react/modules/storage/ReactDatabaseSupplier.java
By the Android documentation, the databases created by the application are saved in private disk space that’s associated application, so it is secure.
Just like files that you save on the device’s internal storage,
Android stores your database in private disk space that’s associated
application. Your data is secure, because by default this area is not
accessible to other applications.
iOS
In iOS the AsyncStorage values are saved in serialized dictionary files. Those files are saved in the application NSDocumentDirectory. In iOS all applications live in their own sandbox, so all files of one application are secured, they cannot be accessed by the other applications.
The code in iOS that handles the AsyncStorage module can be found here: https://github.com/facebook/react-native/blob/master/React/Modules/RCTAsyncLocalStorage.m
And as we can see here the files used to store the values saved by the AsyncStorage are saved under the NSDocumentDirectory (inside the application sandbox environment).
Every App Is an Island An iOS app’s interactions with the file system
are limited mostly to the directories inside the app’s sandbox. During
installation of a new app, the installer creates a number of
containers for the app. Each container has a specific role. The bundle
container holds the app’s bundle, whereas the data container holds
data for both the application and the user. The data container is
further divided into a number of directories that the app can use to
sort and organize its data. The app may also request access to
additional containers—for example, the iCloud container—at runtime.
Conclusion
It is safe to use AsyncStorage to save user tokens, since they are saved under a secure context.
Please note that this is only true for Android devices without root and for iOS devices without jailbreak. Please also note that if the attacker has physical access to the device and the device is not protected. He can connect the device to the mac laptop and extract the documents directory and see all the contents saved under the documents directory.