They just modified the design. You still have the add button, but it’s under Site Search portion of Search Engine. You can see example where I just recently answered this for someone else over at Can't add search engine - #3 by Saoiray
I’m not sure which way to read this. Are you asking if it’s possible to share your search engines between different instances of Brave? If so, I know you can sync your different browsers together and all the settings should sync together (as long as you have selected it to be synced).
The search engine I used as a test the on this browser is appearing on Nightly as well. So I’m assuming should definitely sync with no problems.
However, I’m not aware to import/export things like search engine settings. If that’s what you need, I’ll have to try to tag in people who know a bit more.