First thing: what is a Thunderbird Profile ?

In Thunderbird, the profile stores two main sets of items. First, it stores your local mail, and possibly copies of messages that reside on the mail server (depending on your account configuration). Second, it stores any changes you make while using Thunderbird (for example, changes to account settings and changes to the toolbar).

-- Source.

Finding your Profiles folder

If you need help with that, that article is useful.

Putting together a (bash) script

Then, you can create a script somewhere, that you will execute when you want to backup your Thunderbird profile.

I'm on Windows, so I'm going to use the $APPDATA environment variable that is available in the terminal. I'm using Git Bash's terminal.

This is the script I have done (named backup-thunderbird.sh):

date=$(date -d "today" +"%Y%m%d%H%M")
cp -R $HOME/AppData/Roaming/Thunderbird/Profiles //NAS/Backups/Thunderbird_Profile_Backups/Profile_$date

And I execute it like so (close thunderbird before doing so. otherwise it won't work):

$ sh backup-thunderbird.sh

I use a special folder in my NAS server to store the profile backups. But it could be anything you want, like a Dropbox folder, for example.