The same goes for vinyl shower curtains. 
You know that “plastic-y” smell when you open it?
That is the off-gassing of toxic chemicals you’re smelling.

In this modern world we live in, we are constantly surrounded by toxic chemicals.
But there are simple ways to limit your exposure, and this is one of them.
Buy a non toxic bath mat and shower curtain.  Below are a few options!

Hemp shower curtain-  These are pricey, but apparently well worth it.
Hemp is a strong and durable fabric, is water-resistant (ships sails are often of hemp canvas),
and is naturally resistant to bacteria and fungi. These will air-dry quickly in a well ventilated bathroom,
and can be used alone or as a liner.

EVA shower curtain- This shower curtain liner is made of EVA,
which is a type of vinyl that does not contain chlorine.  supposedly,
it is non toxic and does not off-gas like PVC.
I still would prefer a natural material, like hemp,
but this would be a good option if budget is an issue.

100% natural rubber bath mat- These are non vinyl,
non chlorine, non-phthalate, made from 100% natural rubber.
Rubber will have a smell to it, but rest assured it isn’t toxic.
It is also machine washable.

TPE bath mat- This bath mat is made from TPE (Thermo Plastic Elastomer),
which is a type of plastic that supposedly does not leach toxins and is safe.
Again, I would prefer the all natural rubber option,
but this is definitely better than using a vinyl mat