DMG p. 141 states the following (bolded for emphasis):
A magic item, such as certain staffs, may require you to use your own spellcasting ability when you cast spell from the item. If you have more than one spellcasting ability, you choose which one to use with the item. If you don't have a spellcasting ability-perhaps you're a rogue with the Use Magic Device feature- your spellcasting ability modifier is +0 for the item, and your proficiency bonus does apply.
Staff of Power states the following (bolded for emphasis):
While holding this staff, you can use an action to expend 1 or more of its charges to cast one of the following spells from it, using your spell save DC and spell attack bonus: . . .
If you are a Tiefling Wizard, a Githzerai Sorcerer, a class who took the Magic Initiate or Spell Sniper feats, an Ancestral Guardian Barbarian, a Monster Hunter Fighter, or have any feature which grants you a spellcasting ability outside of the Spellcasting class feature, can you use the spellcasting abilities granted by these races/feats/classes when selecting your spell save DC and spell attack bonus for a Staff of Power (and other similar items such as Staff of the Woodlands, Voyager Staff, etc.)?
Tiefling (Infernal Legacy):
Charisma is your spellcasting ability for these spells.
Wizard (Spellcasting – Spellcasting Ability):
Intelligence is your spellcasting ability for your wizard spells, [. . .] You use your Intelligence whenever a spell refers to your spellcasting ability.
Githzerai (Githzerai Psionics):
Wisdom is your spellcasting ability for these spells.
Magic Initiate:
Your spellcasting ability for these spells depends on the class you chose: Charisma for bard, sorcerer, or warlock; Wisdom for cleric or druid; or Intelligence for wizard.
Spell Sniper:
Your spellcasting ability for this cantrip depends on the spell list you chose from: Charisma for bard, sorcerer, or warlock; Wisdom for cleric or druid; or Intelligence for wizard.
Ancestral Guardian Barbarian (Consult the Spirits):
Wisdom is your spellcasting ability for these spells.
Monster Hunter Fighter (Hunter's Mysticism):
Wisdom is your spellcasting ability for these spells.
It appears to me that they all use the same wording so, RAW, this should work since Staff of Power lets you select any spellcasting ability that you have if you have multiple, correct?
Best Answer
Yes
Assuming you can attune to the item, which explicitly requires you to be a member of the class (or be a monster with spell slots and using one of those class lists).
However, there is no explicit mechanical relationship between the attunement requirement, and the spellcasting ability referenced. So, as long as you have a single level in one of the relevant classes, you can use a different spellcasting ability (which is, presumably better).
Ask your DM. Because of the class requirement to attune, it is probably intended for the wielder to use their class spellcasting ability with the item, so your DM might want to enforce that. Asking them beforehand (say before deciding who gets the item) will avoid any awkward surprises on both fronts.