The reference to it being the "mate" of Glamdring has been a bit blown out of proportion. I don't recall if it is stated as such elsewhere in the hobbit, but it is mentioned in FOTR, in a reference to Glamdring.
"...girt at his side was the elven sword Glamdring, the mate of Orcrist that lay now upon the breast of Thorin Under the Lonley Mountain".
Sting would not have been mentioned as a mate of either by Tolkien since it is a long knife, not a sword.
What exactly Tolkien meant by 'mate' is unclear, but I think all he meant is that the two swords were found together, were both forged in Gondolin, obviously made it out of Gondolin together, and stayed together op to that point in The Hobbit. Mates. There is nothing in the text that I recall actually linking the blades by the same maker or design. I think that is an assumption people have made. Any of the three could have been made at any point in Gondolin, by any one of several smiths in the city. By inking the Sting and Orcrist design, the film makers are simply implying those two were likely made by the same smith, or at least for the same family.
Since the Goblins clearly recognized both swords as Biter and Beater, the term 'mates' by Tolkien may have been a reference to their assumed use against the Goblins in recent times. Probably wielded by Dwarves in the Battle of Azanulbizar. At some point the wielders lost them when they met their fate at the hands of the the trolls, Bill, Bert, and Tom.
I actually have one of the actor scale Orcrist props at the moment, and it is very clearly styled like Sting, but it also fits the book description on several points.