Tolkien understated a lot of the magic, and I suspect he intended a lot of stuff to be mildly magical without ever stating it. And the nature of magic is questionable, as when he talks about the everyday sort of magic of Hobbits hiding well. I always figured the dwarves' mail coats and helmets were all magical in addition to being made of mithril. And that the Black Arrow was magic too.
The things that were overtly magical, like Bilbo's dagger which glowed, and the legendary items like the two swords they picked out of the troll cave, were just the more powerful ones.
Part of it may be that a magic axe may be difficult to tell from a non-magical axe that was made out of a special material or with a special process. And a non-Wizard can make things of minor magic like the Dwarves and Elves could. So where is the break between a Dwarf working at a forge making a fine axe using secret processes and materials, and instilling it with magic? Especially when Tolkien might consider everything the Dwarf is doing to be a kind of magic.
(Side note: when you look around at the world everything seems pretty magical. Seeds, the Sun, eating and drinking, rust.)
Another part is that few can detect magic, so if some woodsman inherits an axe from his father, who knows whether it's magical or just a nicely engraved and well-made axe?
A third part is that because the world is post-apocalyptic, and ruins abound, and wonderful things exist that were made long ago, and war and tragedy spread these things about, it's not unlikely that the woodsman could come across a magical axe. He uses it here and there, notes that it's more effective and needs little sharpening. Eventually he just uses that instead of normal axes. He's going to treat it as a magical axe regardless of whether it really is or not.