Definitive answer from Roller Coaster Tycoon 2's reverse-engineered source code:
A ride's 'value' ("dollar-equivalent of happiness it provides to peeps") is one of the main factors affecting whether or not a peep decides to pay to get on a ride. (If the ride has no entry fee, peeps don't care about the ride's 'value' at all, and so the ride's age doesn't really matter except for maintenance purposes and "how many guests will come to my park at all" purposes.)
A ride's value gets a big bonus during its first four months after construction, a smaller one during the following eight months, and then starts to decline three more years after that, suddenly losing 25% of its value. It then loses another 25% of its remaining value after a further two years. Every couple years after that, the 'value' declines by another 25% of its remaining value. Each of these 'value' drops results in lower amounts that peeps are willing to pay to go on the ride.
Your question is about duplicated rides. In addition to the age-related value penalties described above, RCT2 applies a further penalty to the ride value of every ride which is currently duplicated inside your park. That is, if you have two different Merry-Go-Rounds in your park at the same time, both of them will have a 25% penalty applied to their "value", after any age-related penalties. (In effect, having a duplicated ride type makes peeps treat every instance of that ride type as though they were several years older than they are, and therefore 25% less valuable to your guests.)
This means that if you're going to build a new instance of a ride in order to get the "this is a new and exciting ride" bonus again, then you definitely want to destroy the original ride, or else you basically lose the bonus due to the "duplicate ride type" penalty, which coincidentally precisely matches the 'value' penalty you get for a ride being four years old.
If you destroy all but one ride of a particular type, the "duplicate ride" penalty will be removed from the ride that remains; the penalty doesn't "stick" to the ride just because the ride used to be a duplicate. As soon as there are no duplicates left in the park, the penalty goes away the next time that ride values are recalculated.