Would someone familiar with both of these religions (e.g. a trader or diplomat) have been able to recognize that they worshipped the same God? I'm not talking about society at large, and I imagine clerics on either side had reasons to ignore the evidence, where perhaps an educated (or worldly) layperson might not?
I answered a question about how crusaders understood Islam, which will hopefully be helpful here.
Basically, some Christians believed Islam was a heretical sect of Christianity, so If Muslims worshipped the same God they were simply doing it wrong. Others believed Islam was a type of paganism and Muslims weren't worshipping the same God - perhaps they worshipped idols, Muhammad, or Greek gods like Apollo. Or maybe Islam was created by Satan and Muslims worshipped the devil?
The best-educated people tended to go with heresy. Islam wasn't really acknowledged as a distinct religion until much later.