I do actually think it is your job. Bear with me...
"I'm allowed to say "could you just put those back on the hangers for me please?"
What happens if they say no? Then who does it? You? In that regard, it is your job.
If you piled them up in a corner, and the manager/owner of the shop said, "what are these clothes doing here" and you said "well, its not my job, its the customers" - what would they say? Would you say in the absence of the customer doing it its your job?
Same for the poster who said in Antwerp they wouldn't take them off you - well, what if the customer refuses? Someone has it in their job description, whether its worded as rehanging items or whether its worded under general upkeep and maintenance of the shop floor.
Sorry, but it is your job, in a sense anyway.
I always rehang items on the hangers. I sometimes don't do it as well as people who work there - I don't have the memory/skill for how its done in a particular shop, and some have their own little ways where they sort of wrap the hangy bits round various bits of the coathanger to ensure they don't slip off and do weird things with trousers to make them display nicely and that kind of thing. So I hang them as I would if I were at home, but pretty much expect that a little tweaking and tidying of my well-meant rehanging is needed anyway. Hell, sometimes they have to probably take them off the hanger and start again! But that is...well, their job...I guess...?