Multifactor resolver always uses default instance

[Somebodybro]

Description

In FIRMultFactorResolver defining an instance that is not default does not allow one to finish mfa sign in because it always finalizes with the default instance.

I expected the mfa to finalize, but what actually happened was I got a "user does not belong to this project" error.

Reproducing the issue

Initialize the default instance of FIRAuth with the info.plist to be some firebase project.

Implement a sign in flow with a different instance and catch the multifactor resolver from it's authentication error.

When you try to finalize the sign in with, it tries to use the wrong instance, which can be verified with a breakpoint or by using
a proxy and observing the wrong api key being used.

Firebase SDK Version

10.9

Xcode Version

14.3

Installation Method

CocoaPods

Firebase Product(s)

Authentication, Messaging

Targeted Platforms

iOS

Relevant Log Output

No response

If using Swift Package Manager, the project's Package.resolved

No response

If using CocoaPods, the project's Podfile.lock

No response

[google-oss-bot]

I couldn't figure out how to label this issue, so I've labeled it for a human to triage. Hang tight.

[Somebodybro]

I created a pull request to address this
#12266