How does Obtainium handle package/apk verification?
It doesn't. They are adding a way to check for hashes against a "trusted database", but afaik not live yet
Keep in mind that this is only needed for first installs
What do you mean "only for first install"? Shouldn't you verify packets at every release?
It doesn't but if you are provided a malicious update to an already installed app android handles the verification and will prevent the installation of the update if it was not signed by the developer. The first installation is only time you usually risk a mitm attack