You use your (Wisdom) Spellcasting Ability, and the cantrip counts as a Cleric cantrip
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 ...
It appears to be a 'No'
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 ...
Yes, this is a cleric spell for you.
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.
Each domain has a ...
I think it's worth emphasizing that Shillelagh allows the caster to use their spellcasting ability, what ever that might be. This spell is not tied to a Wisdom modifier.
A warlock casting this spell would use their Charisma modifier, and a wizard would use their Intelligence.
Granted, non-druids would still need to find a way to access this spell, but ...