bread flour: is it all just branding?
Yes, it would be reading too much.  ADM is a food processing company, and it operates a number of mills.  King Arthur doesn't mill its own flour (and I assume neither does Aldi).  The commodity wheat market in the U.S. tends to move huge amounts of wheat to centralized mills, where it is then packaged (and perhaps branded).

That doesn't mean branding means nothing.  King Arthur, for example, requires very specific standards for its wheat, milling process, and final flour characteristics.  Aldi flour is cheaper partly because it probably doesn't require the kind of close tolerances that King Arthur specifies.

Is it possible that some wheat from the same fields ends up milled at the same place and packaged as different brands?  Yes.  But again, each flour brand will have their own requirements for their flour.  If I buy King Arthur flour, I know that my flour will have protein and ash percentages precisely specified to a tolerance within 0.2%.  (For years, I could find an easy summary of those specs and tolerances on the King Arthur website; I don't know where it is now, but I assume the numbers are available on request.)  Most other flour companies don't provide such numbers, or they specify much wider tolerances.  That likely means their batches perhaps aren't as rigorously tested at the mills and/or the requirements the flour brand demands of farmers producing wheat aren't as rigorous.

In any case, the ADM mill probably processes lots of food from lots of sources for lots of branded goods.  Hence the potential for cross-contamination.

Also, it's important to note that "all-purpose flour" is typically a blend of different wheat varieties.  Those wheat varieties may be grown in different locations and/or in different seasons of the year.  Different flour brands will have various requirements for that blending, so I suppose it's possible for a batch of wheat from some farm of some variety to end up in the AP flour blends for multiple brands.  Or perhaps some wheat that was grown on the farm wasn't up to the King Arthur specs for one reason or another, so it ended up in the Aldi flour, but all the wheat from that farm is under suspicion for contamination or something.  In any case, the final AP flour blends will still be quite different.