My 1959 Crestliner had one from the factory when new
so we have always kept it. The original was the total width
of the outside transom. Over the years, we shortened it to
the "wedge" design which is a more refined look.
With a 3/4" inside transom the outside 3/4" piece is not only cool
and decorative, it brings the total thickness of the transom up to 1-1/2" thick
which makes the motors fit better. (in my opinion).
also, in the vast majority, if you decide to do that to your boat,
ample preservation of the wood and transom metal must be in the forethought.
Of course if you make the inside transom board 1-1/2" thick, which most
people laminate two pieces of plywood together for the full length of the transom,
the outside piece is not necessary. Or, you could put a piece of 1/8" aluminum
plate on the outside just for cosmetics. Again, putting a LOT of thought into
preservation before you marry everything together.
and IMHO, it in no way helps to dampen vibration.