Creatures such as the Deva, Couatl and Oni have an ability which lets them magically polymorph into a humanoid, stating that they keep their statistics (other than ones listed as changing). Does this mean they keep their type (Celestial or Giant, in the above examples), and as such would not be affected by spells such as Charm/Hold Person or Crown of Madness (which specifically list Humanoids as their targets), or would their type become Humanoid for the duration of the transformation, and therefore they would be affected by these spells?
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Almost all shapechangers leave type unchanged. Metallic dragons, though, do change their type.
"Statistic," as a defined term, is spelled out on MM pp.6-11. The second element spelled out is "Type," of which both Celestial/Giant and Humanoid are instances. So which stays?
The Coautl's and Deva's shapechange abilities specify that they retain their statistics except for a few callouts which do not include type. Since their Celestial type is a statistic they retain it and are still Celestial while in humanoid form.
The Oni's shapechange ability specifies that the only statistic to change is size. Since type does not change, the Oni is still Giant while in humanoid form.
For completeness, the Quasit, Yochlol, Imp, Doppelganger, various weres, Mimic, Slaadi, Incubus/Succubus, Vampire, and various Yuan-ti are the other shapechangers in the MM. The Night Hag and Lizardfolk Shaman also change shape (ability) without being shapechangers (tag). All shapechange with similar verbiage to one of the samples above and so all retain their type.
Metallic Dragons' Change Shape ability does change type, as spelled out in their ability description.
Answer: The ability will tell you whether or not the statistics of the shape replace your own. At that point anything that affects a specific type will be determined by the creature's new type (if it changed).
I will provide two creature types below from the Monster Manual that draw upon the RAW information on Type: MM pg. 6 includes Type as a statistic.
Number 1: Change Shape while retaining statistics.
A monster such as Deva (MM pg. 16 - emphasis mine) has the ability Change Shape. This ability explicitly states that:
This very clearly outlines that a Deva retains it's type.
Number 2: Change Shape with a change in statistics.
A Bronze Dragon (MM pg.107 - emphasis mine) does not retain it's type.
The following alteration spells include wording that highlights the statistic swap that occurs when they are cast:
Shapechange (PHB pg. 275-276)
Polymorph (PHB pg. 266):
True Polymorph (PHB pg. 283)