The Elven chain is considered a specific armor and is the magical armor list, however it is completely non-magical, so can you make one that is?
Yes.
Elven chain is mechanically identical to mithral chainmail, except in the proficiency requirement, so unless you are not proficient in medium armor, it is actually better to not buy elven chain.
I don't know of any RAW that specifically says specific armor or weapons cannot be enhanced further, although sometimes figuring out the appropriate price can be difficult, and of course anything magical is subject to GM call.
For any non-PFS game, I can't imagine any GM having a problem with enhancing elven chain.
