One of the random possible runes on a Moonblade (DMG, p. 217) is:

The moonblade functions as a ring of spell storing.

I think there's a rule (at least enforced in Adventurers League, not sure if enforced everywhere) that prevents the use/attunement of two items of the same type (ex: can't benefit from two Tomes of Leadership of Influence for a total +4 to Charisma). But here, one item is a Ring, and another functions as a Ring (without being a Ring proper).

So, considering that rule, could someone still benefit from both the Ring property of a Moonblade and an actual Ring of Spell Storing?

Yes, you can benefit from both

It is true that you can't attune to two exact copies of an item:

A creature can't attune to more than one copy of an item. For example,
a creature can't attune to more than one ring of protection at a time.

Both Moonblade and the Ring of Spell Storing require attunement (though the Ring only requires it for some effects). However, as you pointed out, the Moonblade is not a "copy of" a Ring of Spell Storing.

The moonblade functions as a ring of spell storing.

They are functionally equivalent, but not the same. If they are not the same item with the same name, then they are not copies of each other.

However, the only rules preventing use of the "same type" of magic item only refer to whether a character can physically wear or use them at the same time.

Use common sense to determine whether more than one of a given kind of magic item can be worn. A character can't normally wear more than one pair of footwear, one pair of gloves or gauntlets, one pair of bracers, one suit of armor, one item of headwear, and one cloak. You can make exceptions; a character might be able to wear a circlet under a helmet, for example, or to layer two cloaks.

There are no restrictions in any of the rulebooks about benefiting from more than one magic item that gives a similar effect.

Of course, you are still limited by how many items you can attune to if the items require attunement.

Adventurers League rules also allow it

The rule you are thinking of in Adventurer's League does not come from a general rule preventing items with similar effects from being used, but one preventing one-time-use magic items with permanent effects from being stacked.

Some items are destroyed or
rendered nonmagical when used, but grant
persistent effects (e.g. tome of understanding,
manual of golems). Not only do these items still
count even after they’ve been used, you can never
benefit from more than one such item—no matter
how you come by additional copies.

However, that is the only restriction above and beyond what is listed in the core rules, and does not apply here since the Moonblade is not a one-time-use magic item with a permanent effect.

You can use both

The only rule regarding stacking multiple magic items is this one from the Dungeon Master's Guide:

A creature can't attune to more than one copy of an item. For example, a creature can't attune to more than one ring of protection at a time.

Non-attunement items can stack and certainly you could use two different magic items even though they have the same effect.