I don't think at is incorrect, but I think from is the better choice.
If we were together, and I waved at someone on the street, and you asked, "Do you know him?" then I might reply:
Yes, I know him from work.
(It's unlikely I'd say, "Yes, I know him at work.")
Besides, at work carries the connotation of "at the workplace" ("Can I talk with Martha now?" "No, she's at work.") or "busy at work" ("We must slow down in the construction zone – there are men at work.").
I think Shoe's comment is very astute, in that you don't need the prepositional phrase at all, but let's say the nametag showed a name familiar from somewhere else – say it's the name of a percussion player in your band. Would you say:
The tag showed the familiar name of a percussionist from our band.
The tag showed the familiar name of a percussionist at our band.
(Personally, I think I might use in our band, but from would be acceptable, which is another reason I prefer from over at in your example.)
There are several meanings of the word from, but this discussion has centered on meaning #8 in Macmillan's list:
from: 8. belonging to a particular organization