"Corresponding author" is just a corresponding author, the person the journal will communicate with. It's an administrative role, not one of value or prestige (typically; there may be some exceptions, see footnote); perhaps your subvisor is confused about this, but really I think they are just asking you to be their secretary.
The cover letter should be written on behalf of all the authors, even if a particular author signs it. It seems bizarre to me that someone would correspond with the journal on behalf of the rest of the authors and not be named the corresponding author, since that is exactly what a corresponding author is – if I am the editor, I would like to know who I should write to, so I ask you to tell me this by giving me that person's contact information, and I shall deem that person "corresponding author". If your supervisor will handle the rest of the correspondence, then the cover letter might as well be signed by them. If it's unimportant and lacking prestige to be a corresponding author, it's even less important and further from notable who wrote the text on a cover letter, there is no need (and zero value to you) to have your signature there.
This manuscript is not in my field, but my supervisor wants me to get my name on it anyway
Have you actually contributed materially to this manuscript, besides perhaps minor edits of the text itself? In the case where you haven't, you almost certainly should not be an author in the first place, and the rest is moot.
footnote on corresponding authorship:
In some cases, particularly in fields where order of authorship does not signal contribution, people may use corresponding authorship to indicate the person primarily responsible for the work. Additionally or jointly with this, some journals also differentiate between a "corresponding author" and a "communicating/contact" one (the former listed in the published paper; the latter taking on the administrative role I described), or facilitate a different "corresponding author" during the submission process vs. in the published paper. See some prior Q&A here on the topic:
