Dropbox allows a user to drop a file into a shared folder without having this user to be part of the shared folder. This feature is called file request.
Does Google Drive have a similar feature?
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Google Drive doesn't have a feature like "file request" of DropBox.
The closest built-in feature is to transferring the file ownership but this will not add the file to a folder.
By using Google Apps Script or Google Drive API you could achieve a similar behaviour. You can see an example of a script that would do that here.
Google Classroom works together with Google Drive to manage files. On class creation, a Google Drive folder is created on behalf of the class owner and it's shared with the teacher assistants and with the students.
Teachers could create assignments then the students could submit files adding them as attachments to the assigment. On the assignment submission the ownership of the files is changed so only the teachers could modify the files. Teachers are able to return the files to the students, but in this case (using an assignment as a "file request") this step could be omitted.
To learn more about the assigment flow in Google Classroom see Understanding the assignment flow - Google Classroom Help
This is easily done by creating a quick Google Form with one question as a file upload type. You can set parameters on # of files allowed per upload up to 10, as well as file type and size parameters. See example of form setup This will allow you to have a unique URL for the request that you can share, as well as the ability to create multiple types of file requests for different things that can be filed in separate folders.
Teachers could have unique versions for different grades, classes, etc...
Personally, I have a generic one I use for most things but will spin up a new one anytime there is something Specific I need from multiple sources and will share that new URL out until I have what I need then I will disable it when I no longer need it.
Hope this helps!