New car engines used to have to be driven-slow, and had you change the oil after the first 500 miles before going to a regular oil change schedule. Ones build by custom racing shops still do, AFAIK. There are still
special break-in oil for that first 500 miles. Custom shop engines and old manufactured engines were built well, but not still needed a break-in period.
Today, most new cars require no break-in. My wife works in automotive manufacturing, and says that changes in machining technology and increases in precision now leave the surfaces in a condition that doesn't significantly shed metal particles when you start running it. Improvements in oils and better oil filters probably play a role, too.
Guns are not typically machined to automotive precision levels. That may be hard to believe, looking at gaps and tolerance fillers in car panels, but I'll give you a story from thirty years ago. It's mentioned in the last paragraph under 'Overview' in the Wikipedia entry for
W. Edwards Deming, who taught the Japanese statistical manufacturing after American companies refused to listed to him. A friend of mine who took instruction from his consulting group while he was still alive says they told it in different detail. The example (same one mentioned in the Wikipedia article) was for the Ford Escort. It was called "The World Car" because factories around the world made interchangeable components for it. After it had been on the road a few years, word came up the line that Escort transmissions made in Japan were lasting longer than those made in the U.S. So, Ford engineers tore down a randomly selected U.S. made transmission and their metrology lab measured the parts. In particular, they looked at gear tooth spacing uniformity and eccentricity around the shaft hole. All the U.S. made parts were within tolerances in every regard. Then they tore down a Japanese-made transmission. The fellow operating the gear measuring setup set up a gear, and at first thought something had gone wrong with his equipment because all the error measurements he got were zero. They recalibrated the gear measuring setup and found it to operate properly, so he checked again. Same result. Within the resolution limit of the measuring equipment, the gears were perfect. This meant less vibration and that lead to longer life.
I don't know of any production firearm that comes close to that level of uniformity from piece to piece. It's perfectly possible to make happen in theory, but in practice part of what makes statistical manufacturing work is that when many tens of thousands of identical parts are made you can get enough samples to determine more and more details about exactly when to adjust a machine or change a cutting tools or the machine lube or cutting fluid based on measurements and not a guess or a feel. Aside from war production, runs of guns are not typically as large as the runs the automobile industry makes. Plus, you have to be able to capitalize an expensive, high precision, temperature controlled measuring facility. Most makers just aren't large enough to justify that.
So, there will be break-in. Will you see the effects on target? Sometimes yes and sometimes no. A properly broken in gun may simply be uniformly inaccurate, but it will probably last a bit longer than one that has been allowed to gall or have little bits of metal scratch or scrape its surfaces.
The fellow I learned to fit up 1911's from told me that 500 rounds of hardball usually was plenty of break-in. He expected most feed failures to occur in the first 200 rounds and suggested that if there were any feed failures in the last 200 rounds out of the 500, then there was a problem somewhere.
Rifle barrel break-in for accuracy is a different matter. Handgun barrels are typically too short and expected precision levels enough more generous than it is for rifles, that barrel tuning and the like don't really apply to them. For rifle, Gail McMillan felt break-in of a custom barrel just wasted barrel life. Production guns are, IMHO, a different matter because they are not hand-lapped. Some seem to benefit from break-in in terms of reduced metal fouling, but they don't seem to need nearly as much as some routines that are described. Copper is too soft to significantly abrade or wear steel. It is more akin to a strop for a razor that can remove or bend over the thinnest burrs and machining mark edges on the steel. Typically, I think 10-12 rounds, cleaning between each round to re-expose any coppered surface, gives you all the pressure stropping you are going to get. I base this on using a borescope to look at copper left by each bullet. After 5 or 6 rounds, cleaning between each round, I just don't see much further change. I shoot twice as many rounds as it took to make the last visible improvement on the theory my eye and single-shot samples are limited evaluation tools and smaller traces likely remain that could build up over time. Doubling it makes the odds of that exponentially smaller.
Varmint Al just runs a patch loaded with Flitz for 50 strokes through his barrels and calls them broken-in. If they still foul too much, he goes for 50 strokes of JB Bore compound followed by Flitz, IIRC. So that's another deburring approach you can take. None of these polishing agents can remove a measurable amount of diameter off steel.