I need to download ms authenticator on my iphone.

As MrHoffman notes:

iOS 15 - Apple

Passwords

Built-in authenticator

Generate verification codes needed for additional sign-in security. If a site offers two-factor authentication, you can set up verification codes under Passwords in Settings — no need to download an additional app. Once set up, verification codes autofill when you sign in to the site.

If your iPhone can run iOS 14 it can run 15.

Click on a link below for full instructions about how to update with wi-fi or 5G (see: Use 5G with your iPhone

https://support.apple.com/HT211828), or using a computer if you have storage or connectivity issues.

Update your iPhone, iPad, or iPod touch. Click here --> https://support.apple.com/HT204204

Or this video:

"How to update your iPhone or iPad" click here --> https://youtu.be/csJAEyvvruY

Update software on iPhone, iPad, or iPod touch in iTunes on PC --> https://support.apple.com/guide/itunes/update-device-software-itns3235/windows

iOS 15 also includes a built-in authenticator, as a potential alternative to an app for that.

Try entering your Apple ID email address and see if it then prompts for a password. If this works, please let me know.

The resolution to the 'review' issue is often making sure you have entered a credit card or other valid method of payment* even if you don't plan on needing to use it. The document 'If you can't remove your last payment method or use no payment method with your Apple ID' ( https://support.apple.com/HT203905 ) mentions, "if you're using an existing Apple ID with the App Store for the first time, you must provide payment information."

If there is an option to select "Review" or "Continue", select that and enter valid payment information if required. If there's problems doing this:

On a mobile device tap: Settings > [your name at the top] > Media & Purchases > View Account > Review Apple ID.

Alternatively this document explains how to 'Change, add, [review,] or remove Apple ID payment methods' on an Apple mobile device, a Mac, or a PC; click here --> https://support.apple.com/HT201266

More things to try in this 2018 post by Urquhart1244 marked Apple Recommended:

https://discussions.apple.com/thread/8434425?answerId=33542414022#33542414022

If you continue to have problems contact Official Apple Support - Select your region from the list and then your country from the pop-up window. If you do not see your country, see if there is an "other" listing for that region.

https://getsupport.apple.com/?caller=cups&PGF=PGF63005

*Payment methods that you can use with your Apple ID - https://support.apple.com/HT202631

I suggest you also send feedback to Apple about the looping issue. I see this posted at least daily and while there is a workaround, it's clearly a feature that is not working for everybody, or needs better explaining.