This is "Parking Spaces" (1985) by Henry Melton, published in Analog, September 1985. It definitely matches your question, but you have a few details slightly off.
A bunch of unemployed Artificial Intelligence researchers end up at a car company and sneak a bunch of AI into the control system.
Actually there's no real background on the computer team, just that they installed AI into cars to make them run better:
And then, before CAMU [Central Automotive Monitor Unit] finished and went to sleep itself, CAMU truly woke up.
It was strategy time. For most of its day, CAMU was a controller, no more sophisticated than ECU, driven by interrupts and canned programs. But strategy time was different.
CAMU was designed by a computer team who were doing much more than make better engine controllers. They were into general purpose computers, robotics, fifth-generation artificial intelligence—any place the semiconductor technology would let them go. They had just completed a dandy artificial intelligence computer on a chip. CAMU seemed an interesting way to put it to use.
Two cars start communicating with each other.:
The signal had to be a repeating dump of the main ROM from CAMU, a different CAMU.
This was a radically wild conjecture, but what if another system existed? An external car. What if this car's CAMU was different, just slightly? What if this other car was pulsing a control line with its ROM pattern? Did crosstalk work with externals? How could that be tested?
The only thing they have to say is their log files of everything that ever happened to them. Once they exchange 'life stories', there's nothing left to say.
CAMU was ready. It wanted data, and there was one sure way to indicate just what data it wanted. It began dumping the contents of its log file. It repeated the file a total of sixteen times for redundancy.
The other CAMU was silent for a long thirty-eight seconds, then its transmission began. When the signal fell silent, neither attempted to send anything else. They had already told each other everything they knew.
Both cars figure out that the 'driver' of the car is an external component and each car has several.
Each car knew it had a driver; the revelation was that different drivers had different habits.
But when the digestion was complete, so many of CAMU’s conjectures were altered. Other cars existed. And more interestingly, different types of drivers existed. When all the externals were correlated, and a driver composite postulated for each car, the differences were striking.
One car likes his drivers and optimizes performance for each of them.
The concept of different drivers was interesting. As a unifying structure for most of the external inputs, it clarified many of the puzzling irregularities in the data. CAMU now realized that there were three separate drivers associated with its car system. All this time it had been trying to be the ideal car for a composite driver, and never being right for any of the three.
The other doesn't like his and quits working in the middle of an intersection with the horn blaring and the lights flashing.
This isn't quite right; it's more that the car tries harder and harder to get the owners attention to deal with its issues, and he finally gives up on it:
"Oh, the engine would miss every time I tried to give it the gas. I couldn't get any power out of it at all. And towards the last, the dash indicators started giving nonsense readings. The service man couldn't find anything wrong, but I couldn't take the way it was acting. I just decided to trade. Let the next guy live with it."