In wondering if the vinegar solution dissolved a polish and it’s the residue you can see? Do you know if the floor had those polish products applied to it before?
Or maybe it’s the soda or detergent residue. If me, I’d try re-wash an area with just vinegar/water and see if that makes a difference. OR re-wash with hot water and drop of detergent. I can’t decide! Lol

Goodluck! What a pain it’s come up like this.

If it is a polish, for linoleum it’s most often soem sort of soap-wax, there are polish-removers. It would make sense that sth like that would be hard to scrub off, since it’s a protective coating. I’d try with sth like jersey suggests, but I’d start with just baking-soda, lukewarm water, a drop of dish-washing liquid and clean a smaller area and let it dry to see if there’s a visible difference. Baking-soda is excellent for removing fatty stuff (like wax and soap) and it’s mild, it’s used to clean sculptures and statues outdoors, artefacts that musn’t be damaged by detergents.

That is odd. I have linoleum flooring and use white vinegar often to clean it and never had a problem. I’ve even used baking soda + vinegar to try and get rid of stains. Its possible that the film is residue that didn’t get washed off??? I used baking soda to scrub my oven and if I don’t rinse it well the baking soda will leave a film.