Gloves add another d10 damage +dex modifier reduction
The Monk ability is class feature, not a spell
A single reaction is used to deflect or catch a missile. Not two missiles, one missile. The following is a rules-as-written approach, which includes the DMG errata. It is also informed by a magical item amplifying a Monk's ability to long jump using a class feature: Step of the Wind.
If the PC does not have the Monk class feature, then using the magical effect from the gloves will allow them to use a reaction to catch / deflect a ranged attack but (1) does not get the monk's level bonus, and (2) there is no option to throw it back.
All Monks gain this class feature (not spell) at level 3. The only magical effect from this ability, the only ki powered effect, is that "throw it back as an attack" which costs one ki point. The Monk's base deflect missiles ability is not subject to the "no combined magical effects" rule.
Combining Magical Effects
The effects of different spells add together while the durations of those spells overlap. The effects of the same spell cast multiple times don’t combine, however. Instead, the most potent effect—such as the highest bonus—from those castings applies while their durations overlap. Or the most recent effect applies if the castings are equally potent and their durations overlap. For example, if two clerics cast bless on the same target, that character gains the spell’s benefit only once; he or she doesn’t get to roll two bonus dice. (Basic Rules, p. 85)
In the special case of the Monk - specific beats general - the gloves enhance the base class feature just as the ring of jumping enhances the Monk's ability to jump farther when used in conjunction with class feature Step of the Wind. The gloves cannot add the monk's class level to damage reduction since that is not a feature of the glove's magical effect, though the dexterity modifier can be doubled. The two cases where I find ability modifiers being doubled are paladin and sorcerer: the 7th aura for the paladin being based on Charisma (and thus "doubling up" on any charisma saving throws), and the Sorcerer's charisma bonus that adds on top of a bonus to the cantrip green flame blade's damage.
The Monk's ability to deflect / catch a missiles can be improved by a magical item just as the ability to make a saving throw is enhanced by the ring of protection: you are already making a saving throw: the RoP is a magic item that enhances what the PC is already doing.
Mechanically, this means that the monk combines the damage reduction roll (+ level modifier +dexterity) and the damage reduction from the gloves (with no level modifier + dexterity) for the single ranged attack that this applies to. It can only apply to one such attack per reaction.
Bottom line
The monk adds the damage reduction (d10 + dex mod) from the gloves to the damage reduction of that Monk class feature.
Addressing Objections
The DMG errata Combining Game Effects (p. 252).
Different game features can affect a target at the same time. But
when two or more game features have the same name, only the effects
of one of them—the most potent one—apply while the durations of the
effects overlap. For example, if a target is ignited by a fire
elemental’s Fire Form trait, the ongoing fire damage doesn’t increase
if the burning target is subjected to that trait again. Game features
include spells, class features, feats, racial traits, monster
abilities, and magic items. See the related rule in the “Combining
Magical Effects” section of chapter 10 in the Player’s Handbook.
The game features are not the same. One has a level based bonus, the other does not. There is no "duration" listed for either one (neither one is a spell). Reaction is when it takes place during a round of combat (typically, another creature's turn); it is not a duration.
The two methods do not have the same name. Missile Snaring is not the same name as Deflect Missiles.
Should this stack?
Maybe yes, maybe no. Based on the rules as written, nothing prevents this magic item improving on the monk's ability. It's up to the DM to rule on that. Some DM's will take the approach outlined above, and others will feel that the effect is similar enough to be the same, thus the ought to be treated as the same so that the two should not combine.
Choose the approach that fits your table best, if you are the DM. If you are the player, discuss this with your DM, with the points above presented for consideration. Accept the ruling, and play on.
Remember: this item is costing the Monk an attunement slot. The sum total damage reduction for one ranged attack that is added is an additional 1d10 + dex mod (5.5 + dex mod) points average damage. As the monk's level goes up, and 1 point per level is added to this, the influence of the monks level tends to dominate.
Rules as Fun
Boosting avoided damage from a single missile attack that the monk can already boost damage reduction on is not game breaking. It's "Monk does cool Monk stuff a little bit better." See also Great Weapon Master and Barbarians using two handed weapons. Barbarian does cool Barbarian stuff better. :) If the Monk is going to use up one of three limited attunement slots for this, then good for the Monk. These gloves do nothing about melee attacks, spell attacks, breath weapons, and a whole host of other hazards.
