This answer comes from the Intercom Community forum. 👌

Auto tagging capabilities? Is this possible in Intercom, and what are the entryways for those automatic tags to be applied?

Just jumping in here to summarise the thread and provide some tips on tagging users and conversations.

To automatically tag users, create a Series. It's possible to add action blocks in a Series to automatically tag or untag customers when they match your criteria.

To automatically tag conversations, create an Inbox Rule to tag conversations that match your chosen criteria.

See the full thread with all the discussion here in Intercom Community.