"I think it is up to
the person making the claims to prove them.

Not exactly. A scientific 'claim' that is published in a peer-reviewed
journal does require some supporting evidence, but if I'm simply
saying that cable X sounds cleaner to me than cable Y, no such claim
is being made, simply because it is not possible to have access to
another's sensory.

Which is why tests are done on several people to see if the individual
perception is the product of their brain or can be demonstrated
independent of their report of the experience. Using listening alone,
this is how the benchmark for detecting no differences in amps and wire
was established. Then when each new individual reports some perception,
their report can be tested using the same as that for the benchmark and
conclusions drawn accordingly. This same approach is used for such
things as esp, astrology, etc. where that which is reported can not be
experienced by another and tests are established to exclude the source of
that report as something independent of the person's brain. As to
providing support for claims, because the claims often originate in
marketing, it is they who have the burden of providing support in some
established way that can be independently confirmed. Else it is but the
next clever notion of some marketing guy.