Yes, you can use
"user groups" version of the script for this.
For your example you want 3 different user groups: A, B, and C. Each group will be listed in separate file, like
groupA.php - users list for A
groupB.php - users list for B
groupC.php - users list for C
loginA.php - password check/login form for A (copied from password_protect.php, it will include groupA.php as users list)
loginB.php - password check/login form for B (copied from password_protect.php, it will include groupB.php as users list)
loginC.php - password check/login form for C (copied from password_protect.php, it will include groupC.php as users list)
/home - main login page.You'll need to have login form (for example loginAll.php, copied from password_protect.php) including all user lists (groupA.php, groupB.php, groupC.php).
So users from all groups will be able to login here.
/home/A - user selects A from dropdown and password and is directed to this site. this password should only be valid for this section of the website.This area will be protected by loginA.php. If user logged in on /home page, then password will not be asked again. But if user access this site area directly then user will be asked to login.
/home/B - same as above if user selects B from the dropdown.This area will be protected by loginB.php.
/home/CThis area will be protected by loginC.php.
Note: if user logged in for A, one will not be able to access B, and C.