possible for a home to have 2 separate septic systems?

I have an old house in florida built in the 60s. smaller ranch house that is only 1400sf. The 2 bathrooms are on one end of the house and the kitchen and laundry room are on the other side of the house. I know for a fact my septic and drain field is on the side of my house with the bathrooms. However I had a clog in my kitchen sink and after snaking it I believe the old cast iron pipe beneath it got a hole knocked into it as right outside my kitchen window water was pooling up when I used the sink or dishwasher. I dug the ground out and found a pipe leading away from the house but literally towards the yard on the opposite side of the septic tank. The leak is coming out pretty bad under the slab. however I ran my bathtubs for 20 minutes and the washer and never had a single drop hit the ground from under the sink but when I turn the sink on it immediately starts gushing on it. Is it possible for whatever reason that the kitchen sink was literally put on a separate septic tank??? i guess I need to start digging farther into the yard to see where that pipe leads but it seems rather cost prohibitive to have put a septic tank just for a sink when there is one for the rest of the house. Any input would be appreciative if this is something normal i just never knew about, in either case I have some serious problems lol.

1960's plumbing in Florida could be very much Wild West style. I have seen outfalls from septic tanks sent to drainage ditches, laundries/sinks sent to the yard, etc.

Only way to know what you have going on is to have it camera inspected and located, or just dig and follow the pipe.

Sounds like a separate dry well arrangement as was used in my house, as well as many others in the 70's.

If it is, it's purpose is to not overload or hamper the effectiveness of a septic tank by introducing what is referred to as Grey water(non septic). Such as from high grease producing fixtures like kitchen sinks and soap grease from washing machines.
Trouble is they do tend to build up the grease around what's meant to be the leaching media. And block up eventually. In my case, a tank constructed of concrete blocks laying on their sides, with stone surrounding it for leaching into the surround soils.
They were made illegal in my state some time back but I see Texas, Florida, and others are now allowing grey water disposal separate from septic systems.
If you google grey water disposal options in Florida, you'll probably find more info on it.
You may also check with some people in the septic business as they should know whether dry wells were commonly used when your house was built. Or just check with some old timers, like myself.

Like @phishfood says, likely goes to disposal in the yard, or home made dry well.

In my case once it got to backing up too much, I rerouted it to my septic system as that's the way the new systems are designed. But I always thought of running it out to somewhere else, when I had to take care of it it wasn't legal. Now after reading about Texas , Florida, and other states allowing it o be handled differently, I'll have to check the authorities around here.