I think it's a bit unfair to say that Bilbo was ignorant. With perhaps the exception of Cirdan, they were all as much in the dark as to the origins of the Istari. Only Cirdan, being the oldest and wisest, recognised Gandalf for what he was and treated him with due reverence. As he spent so much time with elves, Gandalf became accepted and valued by them over the centuries, without them necessarily understanding his full purpose.
Apart from those elves who had been to Aman would any of them have recognised a Maia anyway, even without the old man disguise?
Men seemed to think that Gandalf was an elf of some sort. From
UT; The Istari :
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'...he was called among the Men of the North Gandalf, 'the Elf of the Wand'. For they deemed him (though in error, as has been said) to be of Elven-kind...
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To the hobbits he was a wise and magical being. He was their friend but he could also show a more severe, sharper side. He was obviously not a hobbit, an elf or an ordinary man. What else could he be but a wizard?
Apart from his own abilities and talents Gandalf had the added enhancement of Narya. This magnified his 'open and friendly' spirit and made him beloved of those he had come to aid. It is possible that the power of the ring also helped to keep up the facade.