The Death Domain cleric's 1st-level feature, Reaper, does the following (DMG, p. 96):
At 1st level, you learn one necromancy cantrip of your choice from any spell list.
If the Death Domain cleric chooses chill touch from the Sorcerer/Wizard/Warlock spell list, does chill touch become a cleric cantrip that uses Wisdom as the spellcasting ability for that character?
Since the text doesn't explicitly say to count the spell as a Cleric spell, the first thing we need to do is consider the counterpositive: What if we asserted that it wasn't a Cleric Cantrip when learned through this feature?
Well, we couldn't rule that the spell must use Charisma (as though learned by a Warlock or Sorcerer) or Intelligence (as though learned by a Wizard) because the rules don't give us a way to decide which we would use. Chill Touch is not a "Wizard Spell/Warlock Spell/Sorcerer Spell", it's "A Spell that appears on the Spell Lists for Wizards, Warlocks, and Sorcerers". There's no intrinsic property of a spell that makes it a "[X-class] Spell", that terminology is just colloquial jargon (and not formally defined at all) for "spells you have learned as a [X-class]".
Furthermore, the Cleric class doesn't give us a way to use Intelligence or Charisma as our Spellcasting modifier. These are properties of the Wizard and Warlock (and Sorcerer) classes, levels in which the Cleric has (for the purposes of our example, assuming a single-classed character) not taken.
So if a Death Domain Cleric were to learn this spell through a Cleric feature, the fact that it's neither on their class list nor a "Domain Spell" is irrelevant: you learned it as a Cleric, and it doesn't make sense to treat it as anything other than a Cleric Spell. You therefore use your Wisdom ability as your Spellcasting modifier for this spell, same as any other Cleric spell.
The Clerics Domain Spells ability says:
[...] If you have a domain spell that doesn't appear on the cleric spell list it is nonetheless a cleric spell for you
While the Reaper ability does not have that stipulation.
Further Magic Initiate says:
Choose a class [...] You learn 2 cantrips of your choice from that class's spell list.
And Reaper says:
You learn one cantrip from any spell list.
I read these as being similar, or even omitting the "choose a class" part.
Optimally you would want to pick a Druid cantrip for this ability, or a new Cleric one.
Based on all the other class abilities that allow you to learn spells and cantrips from other classes this one appears to be the anomaly. I would rule that it counts as a Cleric spell only because the rest of the game's design points to the fact that it probably should be that way anyway. The rest of the stuff in the DMG (race and class wise) appears to be poorly thought out and not balanced well so that also lends to the interpenetration that it should be a Cleric spell.
From the Cleric Domains class ability
Your choice grants you domain spells and other features when you choose it at 1st level. It also grants you additional ways to use Channel Divinity when you gain that feature at 2nd level, and additional benefits at 6th, 8th, and 17th levels.
Domain Spells Each domain has a list of spells-its domain spells that you gain at the cleric levels noted in the domain description. Once you gain a domain spell, you always have it prepared, and it doesn't count against the number of spells you can prepare each day.
If you have a domain spell that doesn't appear on the cleric spell list, the spell is nonetheless a cleric spell for you.
And from the Death Domain
Reaper
At 1st level, you learn one necromancy cantrip of your choice from any spell list. When you cast a necromancy cantrip that normally targets only one creature, the spell can instead target two creatures within range and within 5 feet of each other.
I would say that this makes it a Domain spell, which by extension makes it a Cleric spell for you, and would then use your Wisdom bonus for casting.
Besides, as @Xirema has pointed out, what stat would you use to cast it, other than Wisdom? It's not intrinsically a "Wizard spell" or a "Warlock spell", so you can't say it just uses Intelligence or Charisma, it's just a spell that happens to be learnable by Warlocks and Wizards regularly.
Unless specified, all spells you learn as [class] are cast using [class' spellcasting ability]. In your case, chill touch becomes a cleric spell and uses your WIS modifier.
