At the very least, monster abilities that share a name with known class features
The MM states "A monster with the Spellcasting class feature has a spellcaster level and spell slots". From this, we know that at the very least, Spellcasting is a class feature. The MM also states "A monster’s type speaks to its fundamental nature. Certain spells, magic items, class features, and other effects in the game interact in special ways with creatures of a particular type." Note that "class features" is plural, and why not just say "spellcasting" if that's the only class feature? From this we might conclude that there are other abilities that count as class features.
I will not speculate which monster features are related to class and which are related to race or other origins - I assume this ambiguity is an area that is intended to be left to the DM's discretion. However, there are some known class features that appear in monster statblocks
We know from the PHB that the Abjurer's "Arcane Ward" is a class feature. The Abjurer statblock in Volo's guide also has an "Arcane Ward" ability. The text of this ability is slightly different from the class description text, but it is reasonable to assume that this is only to save the DM from having to do math. Note that the Spellcasting class feature for monsters also has different text.
This is the same for the other Wizard subclass statblocks in Volo's, who all have an equivalent 2nd or 6th level wizard subclass ability with the same effect and name as the original class description (Conjurer has Benign Transposition, Diviner has Portent, Enchanter has Instinctive Charm, etc.). These statblocks are all clearly modeled after classes, and they have some of the very same abilities, so it would be downright bonkers to claim that these are not class features.
If these abilities that share a name between class description and monster statblock are class features, then it only follows that any monster ability that shares a name and has essentially the same effect as an ability on a class description is a class feature.