There are a couple of things going on here. First, there does seem to be a difference between Windows and Mac versions of InDesign as far as font naming is concerned. The Windows version appears to be much more likely to attach the kind of the font to the name than does the Mac version. The Mac version does it only when two kinds of the same (named) font are mounted. The Windows version seems to do it more often (although this might be because the Windows Open Type version of Times New Roman is actually called Times New Roman (OTF) thereby confusing the issue).
It seems that whenever I get an InDesign file that started life on Windows, my Mac version of InDesign complains that Times New Roman (OTF) is missing, and often that Arial (OTF) is missing.