When sizing or flaring on a single stage press, I can most definitely feel the difference in dimension, thickness and hardness of different makes of brass. This is often so distinct that I have gotten pretty good at ID-ing the manufacturer by feel alone. (Especially true for R-P brass which is about the thinnest and weakest of all, Speer which is often shorter, Herters which is harder etc.)
If the variation is this noticeable, it simply must affect the neck tension and how the case grips the bullet, and thus accuracy,
I sort my brass by major headstamp but still end up with a residual amount of mixed leftovers which are not worth sorting down any further.
I can clearly tell the difference between mixed-lot groups and single headstamp groups which have far fewer fliers and unexplained misses.

This all, of course, is somewhat gun dependent as some guns are not particularly accurate with any ammo. It is, however, evident with higher accuracy guns where ammo deviation matters.