Omnia-- not only does the 2004 Gollancraz edition of the Discworld Companion (which is the copy I have) make that remarkable and totally incorrect statement, but in the process of trying to cross check some things, I've found quite a number of other fascinating items (look at what appears to be the Watch Badge -- p.424 in the middle of the Vetinari discussion--quite different from the "former" badge described in the City Watch entry: p. 438). Since the Companion is theoretically by Terry Pratchett and Stephen Briggs, but actually by Briggs, and since Briggs makes up all sorts of history never mentioned by Pratchett, it seems possible that we can disregard this statement completely.
As to the relationship between Vimes & Vetinari--there is no single answer to this question because the relationship changes over time. When the reader first meets Vetinari & Vimes in GG, apparently Vetinari is vaguely aware of the 3 member (growing to 4 with the addition of Carrot) Night Watch. But in the course of the novel, the relationship changes--because Vimes & Vetinari become much more closely associated (sharing a prison cell) and because Vimes saves Vetinari's life (for the first time).
Actually, as we learn in Night Watch, young Havelock knew the Duke when he was transformed temporarily into Sgt. at Arms John Keel--and recognized then the intelligence and command ability that person possess, though Vetinari had no idea who Vimes really was. In fact, he suggests to his Aunt that she may want Keel removed as possible threat to the change of Patricians she is fomenting. Vetinari does not put the John Keel he knew 30 years before together with the Duke he now knows well until the end of Nightwatch. And by that time the realtionship between these two Guardians of the City has changed and grown from that seen in Guards! Guards!
Initially, Vetinari seems to have a kind of grudging respect for Vimes (at least by the end of the book). But Vetinari opperates on the theory that Vimes is (like everyone and everything else in the City)is nothing but a pawn or a part of a vast machine which he,as Patrician, must keep in balance. That requires him to take perverse actions, such as forbidding Vimes to investigate the murders (by the gun or whoever has it at the time) in MAA, which Vetinari does to make sure that Vimes does in fact investigate. Apparently their relationship at this point is one marked by distrust and lack of real knowledge.
By the time we reach Thud!, the most recent purely Watch book, the relationship is essentially one of mutual respect with the knowledge that each is devoted to the city, and each knows pretty much how to manipulate the other. As Jan and several others have said--they are colleagues.
I would go further, though, and say that while Vimes and Vetnari's relationship at that point is not friendship, it is probably as close to a friendship with another man as either of them is capable of.