In the New Testament the speakers are referred to as speaking Hebrew, even though they were likely speaking Aramaic.
Is this an oversight? Was calling Aramaic Hebrew intentional? If 99% of Jews at the time were illiterate could they possibly confuse the languages?
This is really two questions.
The question in the post is much simpler: Did first century Jews know that Aramaic and Hebrew were different languages? In other words, did they know there was a difference between the language they spoke and another language, be it spoken or written? To this the answer is unquestionably yes. Illiteracy would have no impact on this whatsoever. People are massively aware of even minor differences in the way others speak, and always have been. It's the same idea as babies growing up multilingual who parents fear will mix up the languages. They don't. They might code-mix or code-switch, but they don't have trouble knowing that this is one language and that is another, even if they don't have well formed concepts of what "language" means in an abstract sense.
Not only that, but there's a good chance the average first century Jew who was in any sort of contact with other language communities would have had some understanding of the other languages. Remember that the situation of the average American's confident monolingualism is a historical anomaly. Except in remote areas, people have always tended to be multilingual.
The second part of the question can be if the Bible got it right. Did the translators mess up or fudge things? That happens quite often, so could be a possibility here. Or they simply spoke both languages. Or, as is also common in religious communities throughout history, they had one language for liturgical purposes, and one which was spoken in the community. We have first century artefacts written in Hebrew, but that doesn't mean people were speaking it. But Jesus' "first language" being Aramaic also doesn't mean that he didn't also know Hebrew.
I can't speak to whether Hebrew was alive and well as a daily spoken language in the first century, but I can assure you that the people of the time would have known that the two languages were not the same language.
Ancient people weren't dumb.
/u/gingerkid can probably address what the actual linguistic situation was at the time.