Gabriel, an angel, was initially incapable of recognizing Beelzebub, a demon, after Beez changed their face.
This, despite having known Beez for millennia, having been expecting their arrival, and with Beez wearing pretty much what Beez has been wearing for the last few centuries.
Gabriel isn't an idiot, nor does he need glasses, so the only conclusion to draw here is that angels can't recognize demons without being told who they are looking at by a demon-- likely a result of the demons being considered Lucifer's property who can also act as Satan in their roles as diabolical ministers.
This is also why Crowley didn't identify himself to Sitis in any way that spoke to him having his own identity. He couldn't when on assignment to do the work of Satan for Satan.
The Devil doesn't tell you they're The Devil-- you've got to be able to recognize him.
Beez, after changing their face, had to reintroduce themselves to Gabriel-- had to identify themselves to him by name-- in order for Gabriel to be able to see them there.
I bring this up because Beez, in that scene, had no ulterior motives and wasn't trying to hide their identity. They were honest about who they were with Gabriel and wanted him to see them. But...
...what if another demon wasn't as honest?
What if they wanted an angel to see them as someone else, instead of who they really were?
If only a demon can truly identify a demon to an angel, then a demon who didn't want to say who they truly were-- like, say, Lucifer himself-- would need another demon to identify them incorrectly to an angel or angels because those angels otherwise wouldn't be able to recognize him. After all, angels, by definition, are not familiars of The Devil.
And if the other demon would not help with this deception willingly? Well... that would be when it becomes relevant that this story began by establishing that Satan can take forcibly take control of the demons.
The person with the coffee never actually said who they were. Instead, they forced the one person in the room vulnerable to their possession to incorrectly identify them to the angels.
Michael and the others never stopped to consider that Crowley, in this instance, was not a reliable source... or why it was that none of them could recognize the person who came through the door.
Crowley was a lot more right about what was going on in S2 back in S1:
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