Icloud storage

Only on your devices in the Message app. See the following from Use Messages in iCloud - Apple Support

You can use Messages in iCloud on your iPhone, iPad, iPod touch, Apple Watch, and Mac. For your privacy, Messages in iCloud is end-to-end encrypted, which means you can't view or access Messages online via browser. 

Why iCloud. See the following from Use Messages in iCloud - Apple Support

When you turn on Messages in iCloud, your message history is updated automatically on all of your devices. That way, you always have the same view everywhere you use iMessage. When you delete a message, photo, or conversation on one device, it’s removed from all of your devices. And since all of your attachments are stored in iCloud, you can save space on your device.

The only way I know of to recover deleted messages is the new feature on iPhones with iOS 16. Open the Messages app, tap Edit at top left, tap " Show Recently Deleted.

Also note that erasing your phone won't help. See the following from What does iCloud back up? - Apple Support

2. When you use Messages in iCloud or turn on iCloud Photos, your content is automatically stored in iCloud. That means that they're not included in your iCloud Backup.

There is no way to view any messages in iCloud - see the following from my previous reply (with bold added):

You can use Messages in iCloud on your iPhone, iPad, iPod touch, Apple Watch, and Mac. For your privacy, Messages in iCloud is end-to-end encrypted, which means you can't view or access Messages online via browser.