Not writing allegory (a very specific kind of comparison) doesn't mean nothing in the book is a metaphor. It's willful ignorance to pretend LotR has no ideology or non-narrative (moral, epistemic, metaphysical, etc.) content. Tolkien just doesn't want people to say "... and so the orcs are the Germans."

He very definitely did write at least one very clear allegory in his short story "Leaf by Niggle".

His objection wasn't to "metaphors or whatever", but specifically to the technique of allegory. He was a professional expert in literature; it meant something very clear and specific to him (as did the word 'metaphor'). The full quote is,

"Other arrangements could be devised according to the tastes or views of those who like allegory or topical reference. But I cordially dislike allegory in all its manifestations, and always have done so since I grew old and wary enough to detect its presence. I much prefer history, true or feigned, with its varied applicability to the thought and experience of readers. I think that many confuse 'applicability' with 'allegory'; but the one resides in the freedom of the reader, and the other in the purposed domination of the author.

Elsewhere:

"... the only perfectly consistent allegory is a real life; and the only fully intelligible story is an allegory. And one finds, even in imperfect human 'literature', that the better and more consistent an allegory is the more easily it can be read 'just as a story'; and the better and more closely woven a story is the more easily can those so minded find allegory in it ..."