My understanding was always that Morpheus believed the Oracle to be human until Neo told him otherwise. If he hadn't, his crisis of faith upon learning the truth would make no sense. And as you yourself pointed out, his complete trust in a machine would be out of character.
But that "100 years" discrepancy you pointed out is tricky, it's true. We obviously don't have any definitive answers from the movies to explain this, but two major possibilities are suggested from the canon:
Morpheus's Faith
Morpheus is, we see again and again, driven by his faith. He's too young to have known the last "One" personally, but he has an ironclad conviction that this Messiah is real, even when no one else believes with him. He also trusts the Oracle completely, and his faith is reinforced by her demonstrably true predictions for each person who comes to her.
It's entirely possible that Morpheus found (or was given) an explanation for the Oracle's age that he simply accepted through faith. She clearly had unusual predictive abilities, is it really such a stretch to assume that she is ageless, or just very long-lived? People believe that sort of thing in the real world, with figures like Methuselah.
"The War" vs "the Resistance"
Morpheus is unaware of the Matrix Cycle, so he believes that their struggle is the direct result of the first war between Man and Machine. We know that the Matrix was developed during that war, and that the current version was not the first (that is, there were pre-Matrixes like the "Utopia Matrix" before the city-sim Matrix was developed).
It's possible, I'd say it's even a near certainty, that when Morpheus said "for 100 years we have fought these machines", he was referring to the whole war, including the part that pre-dated the creation of the Matrix.

Depending on the relative timeline, it's entirely possible for the Oracle to have been "with us since the beginning ... of the Resistance" without having been there since the start of the war. Let's say the pre-Matrix war lasted 10 years (we know it was long and bloody), then the Matrix was developed over a period of 10 years, then it took 20 years for the first "One" to be born inside the Matrix and start "free[ing] the first of us" as Morpheus describes it.
In that scenario, the war could be 100 years old, but the resistance would only be 60. Assuming the Oracle started young (we know Greek oracles were often young girls), it's not hard to imagine that she might have been a pal of the first "One", and thus have been there "since the beginningbeginning" even if she was a human of normal lifespan."
This is the version I believe. It obviously isn't "true" in-universe, because Morpheus doesn't know about the Matrix cycle, but as a man who thinks he's fighting a long war against a bitter enemy, of course he'd pick the longest number available to underline his point. "100 years" isn't the length of time that they've been popping red pills, it's the time they've been fighting the Machines entirely. Hell, he could be reaching all the way back to B1-66ER and the first Machine uprisings.