There is nothing in the rules that say you can't take the same Invocation twice. However, there would be no benefit to taking an Invocation twice.

This is the "ridiculous" part of the question, the wording on each Invocation actually limit themselves to confer no benefit if taken twice as most Invocations are worded as:

Invocation Name

You can cast [this spell] once using a warlock spell slot. You can't do so again until you finish a long rest.

or

Invocation Name

You can cast [this spell] at will, without expending a spell slot or material components.

Both wordings confer no benefit to taking it twice because you can still only cast [this spell] once until you finish a long rest (or gain a new cantrip, which, again, there's no use taking it twice). Think of it as similar to the limitations on Extra Attacks which don't stack because the wording is you can attack twice. (Oh, I'm sorry, I thought the feature said EXTRA attack, not TWICE attack. A player of mine actually said this)

For the specific case of Agonizing Blast, when you take it the second time, you add your Charisma modifier to the damage. Sweet! Wait... Can't I already do that? Yes. Again, taking an Invocation twice gives no benefit due to the wording.

An exception to this however, is Book of Ancient Secrets Invocation.

If the rules are indeed interpreted as: You can certainly take the same invocation twice but the second'll be useless, then taking Book of Ancient Secrets twice would be tantamount to saying: You can certainly take BoAS invocation twice, gaining 4 1st-level ritual spells, but you just wasted a precious class feature to gain the ability to cast an additional 2 1st-level ritual spells. Was it worth it?

As written, the bow creates its own ammunition "[w]hen you draw back its string and fire." As such, you cannot fire any other arrows you might happen to have on your person: the one the Moon Bow creates would get in the way.

That said, it would not be game breaking to allow the Moon Bow to fire magical arrows. Yes you will do extra damage but you are expending a magic item and a spell slot to do so.