Hi, @ashblade
One of the first things that I would check is the permission of the /opt/mattermost directory post upgrade as it is common to run it as the root user causing permission changes.
Can you run ls -lAh /opt/mattermost/ and share the output here? By default, it should look like this:
vagrant@mattermost:~$ ls -lAh /opt/mattermost/
total 248K
drwxrwxr-x 2 mattermost mattermost 4.0K Sep 14 12:45 bin
drwxrwxr-x 7 mattermost mattermost 4.0K Oct 1 12:59 client
drwxrwxr-x 2 mattermost mattermost 4.0K Oct 1 12:59 config
drwxrwxr-x 4 mattermost mattermost 4.0K Oct 1 17:05 data
-rw-rw-r-- 1 mattermost mattermost 2.1K Sep 14 12:44 ENTERPRISE-EDITION-LICENSE.txt
drwxrwxr-x 2 mattermost mattermost 4.0K Sep 14 12:44 fonts
drwxrwxr-x 2 mattermost mattermost 4.0K Sep 14 12:44 i18n
drwxrwxr-x 2 mattermost mattermost 4.0K Oct 1 13:00 logs
-rw-rw-r-- 1 mattermost mattermost 193K Sep 14 12:44 NOTICE.txt
drwxr--r-- 3 mattermost mattermost 4.0K Oct 1 13:00 plugins
drwxrwxr-x 2 mattermost mattermost 4.0K Sep 14 12:45 prepackaged_plugins
-rw-rw-r-- 1 mattermost mattermost 5.5K Sep 14 12:44 README.md
drwxrwxr-x 2 mattermost mattermost 4.0K Sep 14 12:44 templates
If not, run the sudo chown -R mattermost:mattermost /opt/mattermost/ command and check again if the custom emojis are still broken.