Unless it is stated and priced higher than comparable boxes, it is most likely no. Can be debated, but I would guess that (again, unless stated) recycled product content is less likely to be acid-free. Acid free can be made from recycled, it's just how it is treated that makes the real difference. There is a lot of acid free paper in use out there, I think less so in cardboard. But a lot of it just depends on what is grown near the manufacturer, and we don't always know the source either in geologic or species manner and now things are shipped all over the world in products that come from all over the world!
The acid comes from fresh wood pulp that is not or minimally treated, wood especially from evergreens (fir, pine, spruce, etc.), what you usually think of as "lumber" has tannin in it. That's part of why when it is fresh cut it has a strong smell and is yellow and both of those fade over time or why cheap typing paper starts white and then turns yellow, it's the tannin and that's what you want to prevent harming your fabric. Most acid free is from cotton or other fibers. fine stationary used to have "rag" content listed, that was the amount of plant fiber as opposed to wood fiber.
The thing we really have to consider is what are our goals, I make blankets that are designed to be machine sewn and machine wash and start falling apart after 20+ years of use. Just regular paper is fine. It's going to hold up well in an appropriate humidity minimal care environment. Note, never ever put cardboard down directly on cement. It will wick up tiny amounts of moisture which will decay, mold, stain or otherwise harm fabrics eventually as well as making it a more desirable area for bugs. I use rolling metal shelves myself.
Modern regular paper/cardboard is designed for at least 100 years. Archival is for longer than that! Even newsprint, the cheapest thinnest stuff ever if you put in a few layers you can cover ground as mulch, leave it out in the sun and rain and it lasts a long time -- it's more the walking on it and the various organisms that have access to it than anything else.
So 20-40 years ago I was more concerned with what would storage in gas emitting plastic containers do, especially since I'm especially sensitive to petro-chemical smells. I think most of us can attest that the fabrics are doing fine.
Likewise, 20 years ago we were concerned with how well Sharpie pens would hold up on fabric. Well, now we have 20 years to show that it is still legible and didn't eat holes in the fabric (or whatever our personal concerns were/are, I haven't had a bad experience yet).
The market I was looking to find for my vintage fabrics was for the conservators, making/repairing period objects. They as a group have a higher standard than I do for what constitutes proper storage. They would expect the archival tags and brass safety pins and "proper" storage.