I can assure you that plastic containers (those that are flexible, that most storage containers are made from.)(not hard containers like yogurt containers) do outgass formaldehyde.
Several years ago, as I started my fabric stash. I stored them in plastic storage containers. I always prewashed my fabric to remove excess dyes. When I opened my container a strong smell was omitted. I immediately developed respiratory distress. My "doctor" said it was formaldehyde outgassing.

Like everyone else here, I've had fabric in plastic boxes, no air holes, for 20+ years. The fabrics are all fine.

The only fabric I store in a plastic bag is hand work that I'm currently doing. I would not use the bags for any long term storage.

So the question is, when you give a quilt as a gift, do you suggest how it should be stored?

No, but I do give washing instructions

Yes, it should be stored in a pillowcase, 100% cotton. Never stored in a plastic bag. Especially plastic from a dry cleaners.

Why is it not a good idea to use a cedar chest?

What fabric I have is in a plastic tote. I put tissue paper with bay leaves on top & keeps it just fine & bug free.
Hubby's grandmother's quilts she made us are in pillowcases & frequently take them out & fluff them & back in the case. These also have bay leaves in them.

Mine is all in plastic tubs. We had a house fire and had the fabric not been in tubs I would have lost it all to smoke damage.

Hi Quilters,,, I have all my fabirc in plastic storage containers
,no problems at all. I found this really nice cloth bags to place my finish quilts in--From Clotilde's
Use those fabric on a Reversible Quilt by Sharon Pederson
Fun and not hard to do

formaldehyde is still used to make/process lots of stuff. it's a real problem when new office buildings go up and get furnished with poly upholstery and carpeting. the fumes are circulated through the building via the air conditioning ducts (god forbid you could open a window) and people get sick and even develop asthma. only fairly recently has it become known what the cause is. the practice has been stopped in many places.