For sudo to accept Touch ID for authentication, you'll need to configure your Pluggable Authentication Modules (PAM) system to accept the Touch ID authentication module.
/etc/pam.d/sudo and add the following to the top of the list:
auth sufficient pam_tid.so
When osascript ‘wants to make changes’, likely because of a ‘with administrator privileges’ modifier, it uses the same authentication as ‘sudo’.
This affects how authentication will work for sudo, not just when using AppleScript. You'll now see the Touch ID prompt when using sudo on the command line, such as in Terminal, or when using other applications which request elevation using sudo.