It is unlikely that your grandfathered downspout-to-sewer allows for you to ADD to the sewer with new construction (such as your planned French drain). Such drainage systems were created back when there was only a single sewer system and not the separate sanitation and stormwater sewers we see today. Most of the single-sewer systems will still discharge into local waterways whenever the system is beyond capacity, such as during heavy rainfall. It is common to grandfather older downspout-to-sewer hookups so homeowners don't have to retrofit their rainwater handling systems, but municipalities would prefer you disconnect your downspout to reduce the load on their sewer system.
Your grandfathered downspout means that all rainwater from your house has to go through the sewage treatment regimen for your area. A second stormwater sewer system would usually just drain into a river because it contains only rainwater, and not blackwater sewage from households.
If your local building or code inspectors find out you added your new French drain output into the sewer, you will likely get fined as well as ordered to re-route your French drain to an acceptable discharge point.
Regulations are related to the U.S. Clean Waters Act of 1972 which trickles down to smaller municipalities and eventually to property owners.