I don't think it'd increase meaningfully

This is effectively having slightly better luck on one die roll. We don't increase monster CR because you manage to roll high a few times, yet effectively this is no better than a guaranteed high roll.

Unless one of the following is true, I would not change CR for the monster at all:

• The monster has a high-damage, single-use ability that it will probably use this effect on. (In which case, change that ability to simply not require an attack roll and base CR on that)
• The monster has extremely high damage, very low accuracy attacks (in which case, update that power to be usable only once and not require an attack roll and base CR on that)

Beyond that, it's not going to make a noticeable difference compared to what the luck of the dice is already doing.

Are there effects that fool the spell Identify, and what are they?

Identify (PHB p.252)

You choose one object that you must touch throughout the casting of the spell. If it is a magic item or some other magic-imbued object, you learn its properties and how to use them, whether it requires attunement to use, and how many charges it has, if any. You learn whether any spells are affecting the item and what they are. If the item was created by a spell, you learn which spell created it.

If you instead touch a creature throughout the casting, you learn what spells, if any, are currently affecting it.

Note that it can be cast upon a creature, which a lot of people don't know as, like I did until I re-read it for this answer, base their knowledge of the spell on older versions that could not.

Effects that may fool an Identify:

Non-detection (PHB p.263)

The target can be a willing creature or a place or an object no larger than 10 feet in any dimension. The target can’t be targeted by any divination magic or perceived through magical scrying sensors.

An Identify spell will return no information on a creature or object with this spell in effect.

It is up to the DM whether the language "cannot be targeted" means that RAW the spell simply fails as it cannot target the object or creature for some reason and thus they may realise that something is up or, in my opinion RAI, it works but the Identify is fooled into thinking there is no magic to target and so returns no information normally as if it was non-magical.

Amulet of Proof Against Detection and Location (DMG p.150)

While wearing this amulet, you are hidden from divination magic. You can't be targeted by such magic or perceived through magical scrying sensors.

This has the same power and issues as Non-Detection above, in that it uses the "can't be targeted" language.

Nystul's Magic Aura (PHB p.263)

You place an illusion on a creature or an object you touch so that divination spells reveal false information about it.

False Aura. You change the way the target appears to spells and magical effects, such as detect magic, that detect magical auras. You can make a nonmagical object appear magical, a magical object appear nonmagical

The wording of the spell means that the Identify is given false information, allowing the caster of the Aura spell to make it appear non-magical if it is magical.

However the reverse is problematic as, RAW, the Aura spell does not provide any false spell effect or magical property information, just the school of magic it radiates. The information provided would therefore be odd and suspect.

Mind Blank (PHB p.259)

Until the spell ends, one willing creature you touch is immune to psychic damage, any effect that would sense its emotions or read its thoughts, divination spells, and the charmed condition. The spell even foils wish spells and spells or effects of similar power used to affect the target’s mind or to gain information about the target.

Divination simply does not get any information on the target so an Identify will not identify any spells running on the target. The language here does not have the "cannot target" stipulation given in Non-Detection so it does not have the issue noted there.

Divine Intervention (PHB p.59)

The DM chooses the nature of the intervention; the effect of any cleric spell or cleric domain spell would be appropriate.

It is within the DM's power to identify the cleric's need as somehow including an effect that would fool an Identify spell (which is most definitely within the power of this, as scaled by the "the effect of any cleric spell or cleric domain spell would be appropriate") and choose an effect appropriately. The DM is not constrained to a cleric/cleric domain spell, that is just given as one appropriate possibility.

And of course Wish (PHB p.288) (if you are willing to suffer the consequences

By simply speaking aloud, you can alter the very foundations of reality in accord with your desires.

You might be able to achieve something beyond the scope of the above examples.

You can duplicate any of the effects above, or you can go for something outside the normal bounds of the spell, e.g. "I Wish that this object appeared in every way to inspection and divination magic as being a flametongue". Though given the power of a Wish and the consequences of casting it this way: "I wish this sword to be immediately and permanently enchanted as a Flametongue" may be a better bet, depending on how evil your DM is! Obviously the second is "more powerful" and therefore more risky...

As a sideways look at this question, it would be possible for the following spell to be used to "fool an Identify spell" in a different manner:

Modify Memory (PHB p.261)

You can permanently eliminate all memory of the event, allow the target to recall the event with perfect clarity and exacting detail, change its memory of the details of the event, or create a memory of some other event.

Instead of making the Identify spell get incorrect information, you could change the caster's memory of its results.