How do I get Thunderbird to recognize I'm only moving it to another computer, but keeping the same email, passwords, etc.

Here are a couple of articles describing how to do that. All your user data, accounts info, messages, preferences, passwords, etc. are contained in your Profile. So, basically, the Profile can be copied to the same relative location on the new computer and things should pick up pretty much where they left off. Move Thunderbird data to a new computer https://support.mozilla.org/en-US/kb/moving-thunderbird-data-to-a-new-computer Profiles - Where Thunderbird stores your messages and other user data https://support.mozilla.org/en-US/kb/profiles-where-thunderbird-stores-user-data