Why are mushrooms offered for sale either packed or unpacked in a supermarket?

Some people like selecting their own produce
- others like the convenience of a pre-packaged unit

The interesting thing is that the unpackaged offerings are
comparatively recent in supermarkets - and in the UK are more likely
in upmarket shops.

In addition:  some people like pre-packaged goods because they dislike
the idea that others may have picked over and handled the items;
others like to do just that, feeling that pre-packaged goods may
include some items of lesser quality.
Unpackaged goods suggest that they come straight from the field, which
may or may not be true, but the presentation of them may give upmarket
customers the impression that they are closer to the source, maybe
reminding them of buying in a country market or that wonderful
vacation in the Toscana.

If a supermarket is supplying both, it is trying to cater to both
types of customers.  (I wonder if they unpackage some of the
mushrooms, i.e, if they all come from the same source.)

I'll bet they all come from the same source.

You are spot on about the 'foreign' origins of handling/selecting
food, in the old days in a greengrocers shop, you tended to get
served.
Similarly market stalls used to be a PITA about people handling stuff
(partly because they used to like slipping some crud into the bottom
of the bag).

Visits abroad, shopping in French markets were a revelation.