The original sentence should be “the specialty of an ophthalmologist is ophthalmology” but it’s an odd sentence. In medicine, a “speciality” is something that requires additional training and focuses on specific ailments or skills. In addition, there can be sub-specialities. So an ophthalmologist is a specialist who does eye surgery, but to do retinal surgery an ophthalmologist would need sub-specialty training. “specialty” is used often in medicine, but it can also be used more generally. His specialty is taking free-kicks. Her speciality is teaching piano to students who want to be professional musicians. His specialty is finding investors.

Yes it's a noun, it means something in which one specializes. In this case, the ophthalmologist specializes in ophthalmology. I hope this helps!

Actually, both of those words are nouns. From the MacMillan Dictionary: Specialty: A particular area of medicine that someone is an expert in. Dr Yuen’s specialty is paediatrics.* (Note that in American English, “specialty” is also used instead of “speciality”. The meanings are quite similar in any event.) *US spelling: pediatrics