Could not figure out why I could not change directories with my file manager and stbasic using exFAT with fatFS. It was actually changing directories but using the f_getcwd() function was only showing "2/".
The reason is this:

Note: In this revision, this function cannot retrieve the current directory path on the exFAT volume.
It always returns the root directory path.
http://elm-chan.org/fsw/ff/doc/getcwd.html

For now to use exFAT for keeping track I will have to keep track of the current working directory manually.