It could be an honest mistake born of ignorance.
Since Men don't know much about elves, they may think the beards are a sign of the great age and wisdom of the Istari. Elves are weird and wild, why shouldn't the oldest and wisest of them have beards? (See below: they actually can!)
Remember, we're used to having access to all the information--those living in Middle-Earth often had only rumor and old stories to based their conjecture on. Since Gandalf did live among the elves for a while (and it's hinted he looked like an elf at least sometimes), this is not an unreasonable conclusion to reach.
"Elf" may be honorary or symbolic or both.
Elves are associated with magic so it makes sense for powerful wizards whose origin is unknown to be associated with elves in the stories and titles given to them by ignorant Men.
Perhaps it's just a symbolic title, using "elf" as a way of saying "old and wise and magic and scary," and nobody ever actually thought Gandalf was literally an elf. (Did anyone think Grima actually had a worm for a tongue, or that Saruman looked like Rainbow Dash?)
An aside about sources, since you're asking on etymological grounds.
The name "Gandalf" comes from the oldest written source of real-life Nordic mythology, the Eddas, where he is described is both an elf and a dwarf. His name really is derived from the words for "magic staff" and "elf." Tolkien may have given him these titles in Middle-Earth as well as an in-joke for students of Nordic mythology. (In Nordic mythology, "elf" basically meant "non-god but non-human magic-user" and "dwarf" often just meant a kind of elf that wielded a particular kind of magic.)
As for elves with beards in canon...
Elves rarely grow beards, but they can in their later lives.
There are a few examples of elves with beards (Círdan, Mahtan, and perhaps Tinfang Gelion). They are all older elves, mostly in the third cycle of their lives, so you could spend a human lifetime with an elf and never see him with stubble if he's not old enough yet. It would also be reasonable to think that older elves might shave to appear younger.
The text saying otherwise may be deliberately wrong.
Most of Tolkien's works are written as in-universe retellings of stories, and have flaws and inaccuracies because of this.
So when the narrator of The Hobbit states that elves can't grow beards, it could be a simplification or a mistake of the narrator. Since elves don't grow beards very often--and can't unless they're pretty old--it'd be reasonable for someone who wasn't extremely familiar with elves to assume none of them can.