People say that all the time, but professional cooks have run triangle tests on backyard/farm eggs vs. store bought eggs and people can't tell the difference. At this point, I don't believe there's a difference in taste. The psychological effects that would lead people to believe that difference exists --- a kind of culinary placebo effect --- are so strong that I just attribute everything to that.
Anecdotally I have regularly switched between store-bought eggs and eggs from my friend's little farm over the last 20+ years, and try as I might, regardless of consumption method, I have yet to taste a difference. I have also asked many friends over the years if they notice any difference and all have agreed with me.
It doesn't matter though, I still prefer my friend's eggs to store-bought ones, I'd rather not support that dirty industry.
I cannot tell the difference between backyard eggs and fancy store bought (organic, free-range) eggs, but I can tell the difference between that set and industrial store bought eggs.
My expectation is that what you're tasting is the difference between a very fresh egg and an older egg; there's no doubt that's real (older eggs aren't even functionally the same as fresh eggs).
I doubt that there can be any difference in the egg whites, but the egg yolks certainly have a composition in fatty substances that varies with the kind of food used for the chicken, which should lead to noticeable differences.
While there are some taste differences in egg yolks, the taste difference in meat, between chicken that ate mostly what they had found themselves in a large area with abundant vegetation, insects and worms, and chicken that had been raised in industrial complexes, is huge.
Yes, the constitution of the chicken's feed matters enough to change the color of the yolk, so much so that organic chicken farms introduce additives to feed specifically to color the yolks. And yolk color may indeed cause you to enjoy the egg differently! But there is apparently no discernible taste difference, once you control for egg age.
Multiple groups have done tests on this. Kenji did a somewhat informal test, where he dyed the eggs so you couldn't discern the farm egg. The Japanese have done expert panel chicken feed tests. It's interesting stuff!
I think people want this "farm egg" thing to be true more than it really is true. People got mad at me on Twitter when I griped about those Vital Farms eggs (don't buy those eggs! they cost more than 2x as much as commodity organic eggs!).
Anecdata also, but I can compare the eggs at home (homegrown) vs. any normal restaurant around and there definitely is a notable difference in looks and taste.
That said, this applies to scrambled or fried eggs.
Omelettes not so much, as seasoning might play quite a big part, and even less with cakes, baked goods, etc. in which eggs are just one more ingredient.
Honestly, does it matter? If raising the chickens that yield your eggs makes your breakfasts more enjoyable, is physical vs psychological causality relevant? The important thing here is enjoyment of our food.
It does not matter, outside of the context of a message board, where it is of grave importance.