Fonts with similar names seem to easily confuse ID and it will incorrectly
report a missing font.

Really? I've not experienced that.

Why does ID show and link my ttf font as a PS MT? If I delete this font
from the windows directory, it indicates it as a missing one (TNR PS MT).

Good question, unfortunately I can't see what is going on at your machine. I am guessing either you have not changed all the fonts in your story, placed eps, placed pdf files to the TNR ttf or the font is missing because you have the story editor set to display with TNR PS MT or something like that. The most important thing of all; does your publication appear to print properly? How about the missing font, does the preflight say exactly where it is missing from?