The granting of trademarks is narrower than that. The protection only applies to products and services that are "identical or 'colourfully' similar to the "registered" products or services".
This is not to say that the holder would still not litigate out of fear of losing their mark, though.
That's for using the same word for different things. You can start a new company called Apple in other industries, no problem. You can not use the Apple computer company's logo as yours, though.
You can probably use Sauron, but you can not literally put up a picture of the Eye of Sauron as your project logo without making it appear that you are operating specifically as that trademark holder. With that picture, they're not just claiming to be "a" Sauron, they're implicitly claiming to be associated with whoever currently holds the rights to those movies.
(I'm only hedging on the "probably" because of the known litigiousness of the Tolkien estate. Normally it shouldn't be a problem.)
This is not to say that the holder would still not litigate out of fear of losing their mark, though.