Requested authentication state persistence isn't retained

### Operating System

windows

### Environment (if applicable)

Chrome 131, Firefox 130

### Firebase SDK Version

10.13.2

### Firebase SDK Product(s)

Auth

### Project Tooling

React app / Vue & Quasar app

### Detailed Problem Description

[This page](https://firebase.google.com/docs/auth/web/auth-state-persistence#overview_of_persistence_behavior) states that calls to `setPersistence()` prior to `signInWithRedirect()` should reapply the requested persistence model at the end of the redirect flow. We are seeing that despite requesting `browserSessionPersistence` before sending the link, and pasting the emailed link directly into the same browser tab, the persistence model is reverting to LOCAL.

This also happens with federated auth providers using `signInWithRedirect()`.

### Steps and code to reproduce issue

E.g. minimal piece of code to send a signin link by email:
```js
const signIn = (data) => {
const options = {
url: window.location.href,
handleCodeInApp: true,
}
console.log('options:', options)
return setPersistence(auth, browserSessionPersistence).then(() => {
console.log('auth:', auth)
console.log('Persistence:', auth.persistenceManager.persistence)
sendSignInLinkToEmail(auth, data.email, options).then(() => {
console.log('sent link')
})
});
};
```
Note the persistence printed to console here is:
```
useAuth.js:23 Persistence: BrowserSessionPersistence {type: 'SESSION', storageRetriever: ƒ}storageRetriever: () => window.sessionStoragetype: "SESSION"storage: (...)[[Prototype]]: BrowserPersistenceClass
```

After pasting the redirect link and re-entering the email address on the test app, this code detects & applies the authentication:
```js
if (isSignInWithEmailLink(auth, window.location.href)) {
await signInWithEmailLink(auth, email, window.location.href)
window.location.replace("/") // Dismiss redirect
} else {
await signIn({ email });
}
```
and the `onAuthStateChanged()` handler:
```js
return onAuthStateChanged(auth, async (userData) => {
console.log("onAuthStateChanged:", userData);
if (userData) {
try {
console.log("Persistence:", userData.auth.persistenceManager.persistence)
setUser(() => userData);
} catch (e) {
// eslint-disable-next-line
console.log(e);
}
} else {
setUser(null);
}
```
logs:
```
FirebaseAuthProvider.jsx:17 Persistence: IndexedDBLocalPersistence {type: 'LOCAL', _shouldAllowMigration: true, listeners: {…}, localCache: {…}, pollTimer: 1, …}
```