OK I know this might start a firestorm but hopefully it will not.
What topic doesn't?
I have been in the crazy hobby more years than I want to count. The one thing over the years that seem to be constant is the good old "gear needs to be broken in to sound it's best". Now I'm a believer that gear sounds better/different when warmed up. But when I hear people say something needs hundreds of hours to break in I'm not sure about that. I tend to think what happens is people get used to the sound after so many hours and it is "broken in" at that point.
I can only think of 3 distinct possibilities why you might think this.
1. Your experience is limited.
2. Your system isn't very resolving.
3. Your listening skills are lacking or you don't necessarily know what to listen for.
BTW, every last one of us is guilty of all 3 of the above to one degree or another.
Or a better question is at what point does one believe the gear is broken in?
I think it would help to break things down a bit. To the best of my knowledge there exists 3 processes to this "broken in" process you mention.
1. Electrical burn-in. This applies to any object in which electric current flows.
2. Mechanical settling-in. This applies primarily to objects being positioned in a resting location.
3. Mechanical break-in. This applies to those objects that make motion. Say an internal combustion engine (ICE).
It is somewhat well-known that an ICE engine in a new vehicle's owner's manual will say something like the ICE requires 1000 hours before the mechanical break-in has matured. But what the manual doesn't state is that the engine is locked into position via the motor mounts and there's an all-important spark to cause combustion. So while all focus is on the engine internal moving parts, there is to one degree or another a mechanical settling-in and electrical burn-in processes occurring simultaneously. Perhaps the mfg'ers' focus is on mechanical break-in because failing to follow the mfg'er's guidelines can lead to engine damage or premature end-of-life.
But to answer your question, for some of us it's not a belief but a repeatable fact.
Say a dealer says the new DAC you bought needs 300 hours to break in.
The dealer may or may not know what he's talking about. But that's potentially true of any of us regarding a given subject, right?
So at what point in the 300 hour process does it happen?
Assuming the dealer speaks with any accuracy, full burn-in may be achieved anywhere from 100 to 400 hours. Why? Because we all have a different understanding of what burn-in is and what the process is and what may alter the process, etc.
But if we break things to to a more basic electrical object like an aftermarket AC plug or fuse, the burn-in time in my limited experience is roughly 53 hours (+- 30 min) of time with electric current flowing through it. And with very few exceptions. The bigger objects including much circuitry and/or power supplies or the longer the object i.e. a 2 metter pair of speaker cables verses a 5 meter pair, the burn-in times will vary. My speakers cables for 23 years have always been around 2 meters and as such full burn-in seems to be around the 7 to 10 days of round-the-clock play time.
IOW, very simple electrical objects are rather easy to predict. But the bigger and/or more complex the electrical object, the less predictable the time. But if new, it will burn-in at some point. In my limited experience, most of the compoents I've owned seem to reach full burn-in status around the 150 - 250 hour time frame. A few exceptions include an Esoteric CDP that took 275 hours and a BMC int. amp with an oversized torroidal transformer that I'm pretty sure I heard a an improvement or two upwards of the 500 hour mark.
But it's also important to note that what we call burn-in ain't just burn-in. Regardless of whatever type of vbiration mgmt methodology one may adhere to and since nobody is practicing true vibration isolation, there is also always a form of mechanical settling-in that is occurring to some extent but we don't acknowledge that part. Kinda' like the ICE engine mechanically breaking-in will include a form of settling-in and burning-in simutaneously. And if the audio object includes moving parts like motors, tone arms, speaker drivers, etc, there is always a form of mechanical break-in that occurs as well,
Heck how does one even remember what the DAC sounded like 100 or 200 hours into the break in process?
Memory really only comes from knowing a particular piece of music intimately well. But it's not the memory so much but rather it's hearing a note perhaps even with an unfamiliar piece of music, maybe the blatt of a trumpet, that just seems a bit more natural than what I'm used to hearing from a playback system. Regardless of our alleged listening skills, our ears work very well at discerning less or more natural sounds. If an object is "burning in" and I happen to be listening to an unfamiliar piece but am suddenly impressed with a note or notes, then I will revert back to pieces of music I am familiar with to confirm what I think I'm hearing.
It's also important to note that a simple electrical object like a plug or fuse usually goes thru a 1-time jump in sonics whereas a larger and/or more complex electrical object can and usually will exhibit several jumps in sonics because it's everything along the electric current flow or path that is going through the "burn-in" process which includes simple as well as more complex objects within.
Maybe my brain is just too old these days to remember specifically what something sounded like after hundreds of hours of listening.
You are not alone as it seems audio memory is GROSSLY overrated for many of us. And it seems you are one of the few that's honest about it. IMO.