Much of what PureOS uses is stock GNOME, so I’m pretty sure you could use most of the same applications on both Manjaro and PureOS just by use Manjaro with GNOME.
PureBrowser is a fork of Firefox with a few add-ons bundled by default and maybe some configuration changes. I doubt you could install PureBrowser easily in Manjaro, but you could recompile from the source (available here: https://repo.pureos.net/pureos/pool/main/p/purebrowser/)
Or just manually change Firefox to effectively turn it into PureBrowser (though I’m not sure if there’s a simple list of the changes from Firefox to PureBrowser)