Simple answer: no.
You don’t have to build an Academy in your second or third city in order to get the bonuses from your researches.
You can test this by building a level 9 Vault in your second city > press the resources tab bar > notice that you have “Protection Limit: 1.80m”. Both in your first and second city if you have level 9 Vaults in both places.
If you have a level 10 Vault in your first city, you should see a 2 million protection there.
The same goes for other researches, training troops speed, resource generators, construction speed etc.