[Auth][desktop] Save user data after reauth and update user info
[Auth][desktop] Save user data after reauth and update user info

PiperOrigin-RevId: 264923297

firebase/firebase-cpp-sdk,firebase/firebase-cpp-sdk,firebase/firebase-cpp-sdk,firebase/firebase-cpp-sdk,firebase/firebase-cpp-sdk,firebase/firebase-cpp-sdk,firebase/firebase-cpp-sdk,firebase/firebase-cpp-sdk