Chrome asked me to choose a new browser, I selected Google, and Brave jumped in ???

Google is not a browser. Google is a company. Google makes Chrome. Google would not ask you to select a new browser if you were already using Chrome. Google is also the name of Google's search engine. The thing you see when you go to https://google.com. Again, not a browser.

Brave is a browser. You can't just magically switch to a new browser. You need to download and install this new browser first before you can even use it. Did you install Brave from the App Store?

What exactly happened?

Where did you select this new browser?

What were you doing?

Where are you now seeing you are using Brave?

You need to explain in detail what you did, and what you now see, before anyone can even begin to advise.

On an iPhone to use a Browser, you just need to open the browser app you want to use.

If you instead opened the Brave search website, that's just a website, not a browser.

Chrome didn’t ask that.

There is simply no way that Google would not want to collect your data, either via the browser, or via the search engine.

That was probably an ad.

The ad would normally have taken you through the app store to load the other web browser.

Brave is a different web browser, also based on Chrome.

If you don’t want to use Brave, delete the app.

If your iPhone 13 is still running iOS 16 as indicated by the footer, also consider an update to iOS 17.5.1; to current.

Here is how to: Update your iPhone or iPad - Apple Support

Here is what to do: If your iPhone or iPad won't update - Apple Support

It appears you may be confusing the Browser and Search Engine. Chrome is the browser and likely the app that you use to open a website and the search engine that Chrome uses can be one of many different search engines, such as Google, Duck Duck Go, Yahoo, Bing, and even Brave has a search engine.