Modern ECU units and transmission computers store an increasing amount of persistent data between starts, which was virtually nonexistent in the 1970's and early 1980's. Usually this can be cleared by disconnecting the battery for an extended period, which is necessary in some rare instances.
For the engine computer:
- recent and long term fuel characteristics: Every time you refuel, there are slight variances, such as additives or lack thereof, ethanol percentage (which requires longer duty cycles), octane, etc. and is typically refined through the use of oxygen sensors which have made great strides in accuracy improvements with wideband technology.
- timing vs. knock parameters: if you fuel your modern premium-only car with regular octane fuel, the car will not blindly predetonate until it breaks, it will 'learn' to pull timing based on inputs from the knock sensor, and occasionally push forward until some knock is discovered.
- various sensor trims: many sensors vary from the factory and over time, and the ECU will sometimes 'trim' them into expected ranges, such as the throttle position sensor (TPS), O2 sensors, fuel level sensor, and many others.
For the transmission computer (automatics only):
- shift points: as various issues affect the engine power output, including sensor trims and driving style, some parameters are stored to assist in shifting at ideal conditions.
(I don't drive or service or care about automatic transmissions, so there are perhaps others?)
In any case, as storage and computation power increase and also continue on a trend of decreasing price, and we now have self-driving vehicles and witlessly controlled vehicles, I expect that more and more parameters will be stored and utilized to optimize the driving experience.
The self-driving is hardly yet what I would call 'AI', but this along with the rest of the learned parameters are veritably non-mythical and every OBDII vehicle (1994+) is performing at least the fuel characteristics and timing calculations based on persistent data storage.