In no sense is the Balrog a dragon, either. It's a demon of fire and shadow. Dragons are "worms", ie, snakes: they have to be reptilian to qualify as "dragons" in all but the very loosest usage of the word.
The reviewer was just confused, because in the scenes where they fall, you can see the fire and wings of the Balrog, but not much of the body. It does look a bit dragon-ish, if you forget what it looked like standing on the bridge.
"Dragon" is, however, a catchall monster term you encounter throughout literature, particularly if it has biblical imagery. Even the devil is "that old dragon." When hubby was discussing the battle falling through the mountain, he referred automatically to the Balrog as a dragon because of the associated fire. I've also seen many old illustrations of dragons that include horns on the head. Now, he knows that a dragon more typically is a lizard-like creature, but the term was one we could both understand.