I would like to apply for several positions at the same company and department. My cover letter uses more or less the same structure for any application: introduction, common aspects between me and the position, closure with invitation to schedule an interview. Well, in 2nd paragraph in particular refers to requirements or tasks of the advertised position (in order to show how I comply with); aside from that the wording is basically the same. I am afraid it could sound a bit unnatural if the HR manager reading them is the same.
I had already a look on this post and related links to find some hint, but the main differences with those cases is that this company has an online system to upload cover letter, resume, etc. to be sent; so, I believe mentioning all the wished positions in once may create a lot of confusion and no sense.