AUR is a single repo where users submit source packages only (PKGBUILDs). When you install a package from AUR, you’re downloading the PKGBUILD and building it yourself.
Copr is a service that allows users to create their own repos. Packages are built on Copr.
Basically AUR is like a family cookbook where you share recipes with each other, but you have to make the food yourself. Copr is a mall where everyone can open their own restaurant serving cooked food, but to get a specific food you have to know which restaurant to order from.
Third-party repos like Brave are not related to Copr (or AUR).
If Brave had their own repo for Arch packages not-real-arch-repo.brave.com, you would also have to know the URL to use their repo on Arch. But nobody outside of Arch makes Arch packages (case in point, Brave makes deb and rpm packages). Hence you have the AUR where users try to package everything.
You can search Copr projects (not packages) from cli. See dnf copr (from dnf-plugins-core which is installed by default). Many Coprs follow the convention of 1 project per package or group of related-packages, but this is not a rule; some users put all their packages into 1 project so the name is not representative of what packages are in there (like a personal mini-AUR).
You probably should not use the cli to find and install packages without checking how they’re made (just like AUR, which advises you to read every PKGBUILD before you build them). User-made packages don’t go through any official review process and there can be some questionable ones.
Yes. Point is: in Arch, someone else has already figured out their latest repo URL, I don’t have to look it up to check it.
Yes. Which is why I’m looking for those “ready recipes” to cook up Brave without having to search for its installation instructions. In Arch, if the installation process is changed for whatever reason, someone else already took care of that and updated the installation script.
If Brave was 100% free, it could be packaged in Fedora (or someone would’ve already packaged it on Copr). But there are restrictions on the signing keys. This is why it’s only on AUR and not Arch repos, according to the AUR packager.