I have a 93 mazda 929. Its rear mufler has a wire connected to a
shutter in one of the exaust pipes that supposedly opens when the car
is running fast. I want to ask what is the shutter for and if it is
safe to change that mufler to an after-market one that does not have
the shutter but just two regular open pipes.

Yes it is safe. That contraption you mention is a way of making the
car run more efficient when at high speeds and quieter at lower
speeds.

Imagine your engine being a huge air pump. The more restrictions it
has in place pumping the air out, the less effecient it will run (more
fuel required to maintain a flow rate under constriction vs. no
constriction).

The trade off is noise produced. At 100mph you are not concerned about
noise from the muffler.

This is why people add bigger "pipes" that change the sound of the
engine - they are messing with the noise/efficiency forumla