Does my Django app have to be the same name as the project directory?
There won't be any issue with having the different name of parent directory and the project directory (the one with settings.py).
You can try renaming the parent directory and it would still work the same way but the name of project directory should not be touched since the same is linked with project settings which are used to run the project.
I've pulled my django repo from bitbucket onto my digital ocean server. My project folder name in my server is project whereas my initial app from my repo (the one with settings.py is called app. Does this cause any problems? Because I know when you create a django project offline, the first directory is always the same as the parent directory.