"On the first of every month" vs. "every first of the month"

The difference is that the first is a prepositional phrase and the second is a noun phrase.

There's nothing illogical about the second one. Nor is there any reason to hyphenate it beyond stylistic preference. It's just a different way to say the thing, but everyone would understand it:

"On the first of every month I like to go out and buy a new hat" "Every first of the month I like to go out and buy a new hat"

It's certainly a less conventional way to say it to my ears, but in terms of speech there's really no wrong answers as long as you're understood. On the other hand, if you're writing an essay I'd stick with the prepositional phrase because it sounds more conventional to me like I said. Even if it's regionally accepted, if you're writing an essay you don't want it to only appeal to one region!

Hope that helps!

"On the first of every month" is a bit more conventional and widely used. "Every first of the month", while also proper English and unambiguous, sounds a bit clumsy to me. I don't think I've seen it very often.