I studied The Classics for a few years, it's not named specifically after Greece and Rome, but it is named after the Classical antiquity period. Egyptology also takes its suffix from Greek Logia, which just means anything from a love of learning to just used as describing a subject - like technology, which itself is a mix of the Greek τέχνη and Logia. It's put on words that describe professions where you study a particular topic, so you have an Egyptologist, but you also have an Archaeologist, an Anthropologist, a Technologist, etc etc etc.

The answer you will be looking for: the University of Copenhagen offers Assyriology both as a Bachelor and as a Master's.

While your friends aren't specifically wrong in the context of what Egypt or other countries may offer in terms of an education in specific -ology topics, there are places in the world where you can take a 5-year degree in -ology that isn't centered around Egypt at all, Assyria being the most popular alternative here.

It's also worth noting that just because a course isn't called -ology, that doesn't mean you don't delve deep enough in to it. We have Bachelor's and Master's that are fully accredited that cover everything from Japan, to Europe, America, China, Germany, Korea, etc. These courses offer the same as an Egyptology Or Assyriology course, in that you become an expert in that culture - the history, language, evolution, and present-day role in society. It's just called Korean Studies rather than Koreanology for instance. If your friends want to mince words, they have a case, but not really, lol. It's basically just a matter of naming conventions, and the etymology behind the suffix.

Hope I helped!