I installed minibb on one domain and replaced the old style css from the original minibb style. I put a few addons into the forum.

So if I want to make a duplicate install of this forum on another domain, can I just do a full ftp download of all the files on the original domain and then upload those to the new domain and do an install from there?

Or do I have to use the original files from the original zip and then do a fresh install and then do all the addons and template file replacements again?

You could merely take all existing files, transfer them to another domain, modifying just setup_options.php and possibly settings for add-ons, which may have them different on each server, f.e. domain setting and file paths.

In this case, you just should install the database manually, if you know how. MiniBB installation provides _install.sql file for manual installation of the database - you should just preferrably replace table names for each server and then also adjust this in the options file.

In the case of moving an existing database, you should make its backup first, then import this backup on another server.
There are tons tools for it, incl. that ones build in cPanel or phpMyAdmin.