The earliest version of this quote, I read in an article or book by Col. Jeff Cooper, and was worded to the effect of "Beware the man with one gun; he may know how to use it." The context of the story was to be careful not to discount the skill of the man who has only one gun, because if he practices with it, he may very well be very skillful. It was NOT used as a persuasive tool for folks to literally own one firearm, and was NOT meant to mean that the one-gun man IS going to be skillful.
On the other hand, one of my mentors in the areas of both shooting and law enforcement urged me to pick one handgun design and get really good with it. It took me a while to take it to heart, but when I concentrated on the 1911, and set aside my DA revolvers, the advice was proven to be true. Using my agency's qual course as a benchmark, I could "qual" Expert just about on demand with a sixgun, but had difficulty attaining that level with a 1911. Well, after several months of leaving the DA sixguns in the safe, and diligently shooting the 1911, I was able to qual Expert with the 1911 just about every time.
OK, did my sixgun skill diminish? Well, in my case at least, no. When some particularly nasty inmates escaped from an area prison, and operated together as a close-knit gang, including the murder of a fellow LEO, I remembered the flat trajectory of my .357 sixguns, and as my chief did not allow us rifles at the time, decided to re-qual with a sixgun, and keep it cased in the patrol car, to augment the 1911 in the duty holster. In spite of not shooting a DA sixgun in over three years, I shot the qual course at a very high level, with the aggregate group being as good as ever. (I had not "grandfathered" the sixgun, so it could only be a "back-up" gun, and the range officers do not score back-up gun quals, except as pass-fail.)
As for getting really good with one weapon system, I think what we seek is unconscious competence, the ability to perform all the functions without conscious thought. This gives us an edge in a time-is-life situation, to borrow a term from Travis Haley of Magpul Dynamics. Folks may pooh-pooh the fast draw, but if the choice is draw against the drop, or die, gentlemen, it is time for fast draw. If the choice is a fast reload, or die, that is no time to figure out an unfamiliar mag release. If every little change in the loop takes three-quarters of a second to recognize and adapt, I want my carry gun to be VERY familiar.
This does not mean one cannot learn to operate other weapon systems. Indeed, I know how to run autos with Beretta/Walther-style hammer-drop safety levers, Glocks, the HK P7, the AK rifle family, the AR15 rifle family, the Garand rifle family, several lever rifle and bolt rifle systems, several single-shot rifles, and have a working knowledge of the M1 Carbine from handling a non-firing example. If I have to run one of these other systems in a fight, however, I may well have to think my way through at least some parts of the weapon manipulation, which may or may not cause a delay that gives an enemy the edge he needs at a moment in time.
All of my serious handguns point well in my hands. I presently choose to carry only long-stroke DA handguns as go-to pistols, that I might draw and fire in an urgent hurry, reflexively. I have other handguns that I might indeed use for fighting, that are DA/SA, or the SA-only 1911, that point well in my hands, but they are "war bag" guns, or something that I pick up proactively, and have the chance to make the mental adjustment to such niceties as to place my the center of the distal pad of my trigger finger onto the 1911 trigger face, rather than the distal crease on the center of the face, as I do with DA handguns.
FWIW, I will not mix Glocks with my carry guns, because things such as stroke length and reset cause me to have to stop and think, plus they sit so low in my hands that my trigger finger drags along the frame, which plays hell with practical long-range accuracy. My agency uses "simunition" Glocks in training, and after a class, I will dry-fire a SIG or SA revolver to reset my brain and reflexes.
I have no problem at all going back and forth from DAK SIGs to Ruger and S&W DA sixguns; indeed, my cheap-practice gun for both platforms is an S&W Model 17 K-frame chambered in .22 LR. Both systems point the same, the trigger stroke is the same, and so no conscious thought is needed to fire the first six shots. Before I run out of ammo, my brain has had enough time to perceive whether a revolver or auto is in my hands; I have never tried to stick a magazine onto a sixgun, or open the cylinder when handling an auto.
The duffer who is not familiar enough with any of his guns to operate them at a level of unconscious competence will not understand what I am saying. The truly gifted man who can run any virtually any weapon with equal ease probably does exist, too, so I am not calling anyone a liar; such folks are blessed and can disregard my advice.
I hope my rambling might make some sense, and be helpful to someone.