Does Beorn know what Gandalf is?
I think you're overthinking the scene. Nothing in that conversation indicates that Beorn knows anything at all about Gandalf, except that he's an intelligent guy who's pretty well-informed about events in Middle-earth. Since he just had the wizard as a dinner-guest, it seems completely characteristic for Beorn to know that much.
It's Gandalf who steers the conversation to Sauron (indirectly), not Beorn; there's absolutely no reason to believe that Beorn has any information about the nature of the Istari or their mission.
In fact, in a canon context, he definitely did not, because practically nobody knew.
Does Beorn know who Gandalf is?
Well he says he doesn't, and he has no reason to lie. All of the points raised (both in the question and in the comments) to suggest that Beorn knows more about Gandalf than he admits is, to my mind, stretching the evidence; there are more reasonable explanations.
Beorn talks about Sauron, which is a niche subject only certain people would be interested in. This is true, but ignores the fact that Beorn only brings up the Necromancer in response to a question Gandalf asked; the conversation literally goes (paraphrased):
Beorn: Orcs will cause you trouble.
Gandalf: Why are they so aggressive lately?
Beorn: Because they're in league with a sorcerer in Dol Guldur
Gandalf: What do you know about him?
Beorn: That he attracts evil things. Oh, and by the way, raises the dead.
Beorn isn't offering up this information, he provides it because Gandalf asked for it.
He asks Gandalf to tell him about Sauron. Again true, but not indication that Beorn knows anything about Gandalf, except that he has an idea what's going on in the world. Watching Gandalf's face during the conversation, it's abundantly clear that he's troubled by what Beorn has to say, so it's not a stretch for Beorn to think to himself "This greybeard knows more than he's letting on."
What's more, Beorn clearly has an interest in news about Sauron, above and beyond any potential interest in Gandalf; he straight-up admits that he remembers (or, at the very least, remembers lore about) Sauron's domination of that land in the Second Age, and he has very unhappy memories about his treatment at the hands of Orcs. If he'd heard rumours of the dead walking around, something deeply associated with Sauron, why wouldn't he want some confirmation?
He draws a distinction between Gandalf and Saruman. This point is raised in relation to the following exchange from the clip:
Beorn: If [Sauron] has returned to Middle-earth, I would have you tell me.
Gandalf: Saruman the White says it's not possible. The Enemy was destroyed, and will never return.
Beorn: And what does Gandalf the Grey say?
The Hobbit: The Desolation of Smaug (2013)
But nothing about this implies that Beorn knows who Saruman is, or what his importance is. You don't need to know any lore about Middle-earth to see that Gandalf is deflecting the question, using this "Saruman" person as an excuse. If you wanted to, you could replace the above dialogue with the below, and the conversation would be exactly the same:
Beorn: Has Sauron returned?
Gandalf: Saruman says no.
Beorn: I'm not asking Saruman, I'm asking you.
The Hobbit: Simple English translation
Why does Beorn trust Gandalf?
I don't recall if the movies touch on this, but in the book Beorn is familiar with (and respects) Radagast (emphasis mine):
"Who are you and what do you want?" [Beorn] asked gruffly, standing in front of them and towering tall above Gandalf.
As for Bilbo he could easily have trotted through his legs without ducking his head to miss the fringe of the man's brown tunic.
"I am Gandalf," said the wizard.
"Never heard of him," growled the man, "And what's this little fellow?" he said, stooping down to frown at the hobbit with his bushy eyebrows.
"That is Mr. Baggins, a hobbit of good family and unimpeachable reputation," said Gandalf. Bilbo bowed. He had no hat to take off, and was painfully conscious of his many missing buttons. "I am a wizard," continued Gandalf. "I have heard of you, if you have not heard of me; but perhaps you have heard of my good cousin Radagast who lives near the Southern borders of Mirkwood?"
"Yes; not a bad fellow as wizards go, I believe. I used to see him now and again," said Beorn. "Well, now I know who you are, or who you say you are. What do you want?"
The Hobbit Chapter 7: "Queer Lodgings"
Since Beorn evidently has a measure of respect for Radagast, it stands to reason that he would also hold a measure of respect for one of his associates.