Can I use the exact same research proposal for different scholarships?
No problem, often people in your situation do the same. Even professors try to improve failed grant proposals in order to submit it again in other calls. 

In your case, I think you must send the same project, since you have already delineated the work you will perform with your potential supervisor and you are just applying for funding. After the results, if both get approved, you will need to choose only one of the grants. 

Plagiarism is a completely different thing:


  the practice of taking someone else's work or ideas and passing them off as one's own.


In any case, I think you should talk to your supervisor before submitting.