Here is how to set Brave (or any other standard interface search engine) as the default search engine in the Chrome browser on Android mobile devices. This may work on Chrome for iOS as well, but I have no such device on which to test it.
When running Chrome, select the Settings page by tapping the “⋮” menu at the top right and selecting “Settings”. On the Settings page, tap the “Search engine” item to show the Search engine page. This will show a list of search engines with the current default checked. If Brave, which will show as “search.brave.com” appears, just check it and you’re done.
However, since Brave is not among the standard list of search engines, proceed as follows. Go back to the main browser window and launch the Brave browser from the explicit URL:
Now do a search for anything at all. When the search results come back, navigate back to the Settings/Search page. You should now see “search.brave.com” in the list of search engines. Select it and it should now be the default browser. To be sure, try a search from the main browser address bar and confirm that Brave displays the search results.