My vet tech partner says: yeah it’s the middle finger. They have splint bones on either side which are remnants from the other fingers.

Yes it is the middle finger/toe.

http://rspb.royalsocietypublishing.org/content/284/1861/20171174

Horses have 5 fingers/toes in their genetic, but only the middle ones develop. The reason is that one finger/toe has advantages for a huge weight and wanting to run quite fast.

If this paper is anything to go by: yes, the hoof is the “middle” finger, with difficult to recognise vestiges of digits to the left and right still present.

Wiki says yes: https://en.m.wikipedia.org/wiki/Horse_hoof

“It is a complex structure surrounding the distal phalanx of the 3rd digit (digit III of the basic pentadactyl limb of vertebrates, evolved into a single weight-bearing digit in horses) of each of the four limbs, which is covered by soft tissue and keratinised (cornified) matter.”