Yes, this one was added to the alternatives. Thank you for your comment.
roughly like "ee-thee-ee" (with the "thee" pronounced as in the English word).
ίδιοι has a vowel followed by a glide followed by a vowel. i, then y, then i. think of how some people (especially in america i think) pronounce israel 'israYel'. Take that 'yuh' sound and add an i at the end : 'yee'.
you end up with ee-thee-yee, although the second i is often elided/omitted when spoken, and you get ee-thyee
This sound is not written though. It is different saying ίδ-ι-ος, specially in Ancient word meaning the identity, the characteristic than saying ίδιος in Modern Greek, with this pronounciation as above, meaning the similarity with.
Well, "identical" means "exactly the same" and the Greek word for it is "ολόιδιος" (όλος=whole, entire) and it's even more emphatic concerning the details. I think it would be better to leave it as it is.