About the voltage, 13.4 volts sounds like a correct float voltage for the stage when the battery has already been charged for a long amount of time, for a temperature of approximately 30 degrees Celsius. It's also possible to use 13.4 volts as the main charging voltage for the entire duration of the charge. In that case the battery charges slower than it would with a 14+ volts voltage, but charging that way is safer for the battery long term because it will not be overcharged.
It's possible the car in question doesn't have a current sensor sensing the actual current into the battery (excluding all parts that are immediately consumed by electrical loads and supplied by the alternator). If this is the case, 13.4 volts seems like a good compromise at 30 degrees Celsius temperature. The battery will charge a bit slower but will last a long amount of time because it never is overcharged.
I know some cars use 14+ volts charging voltage continuously. In this case, the car manufacturer probably estimated that standby loads when the car is idle plus the starting motor load when starting to drive take some amount of charge away from the battery, and that in most cases car trips are not so long that 14+ volts would have enough time to do actual damage to the battery. But for a car that would be used for only long road trips, the 14+ volts continuous charging voltage could cause an early battery death due to overcharging.
Ideally, a car would have a sensor measuring exact voltage at the battery terminals, plus a sensor measuring exact current into our out of the battery. Ideally charging would be started with a 14+ volts voltage with a current limitation (bulk/absorption charge), and then when current into the battery falls below a threshold, switch to a float charge stage with about 13.4 volts voltage at 30 degrees Celsius. Ideally there would also be a temperature sensor right at the battery, and voltage temperature compensation would be done so if battery temperature is not 30 degrees Celsius, the float voltage would be changed from 13.4 volts to some other value. Higher temperatures need less voltage.
Note the 30 degrees Celsius refers to the temperature of the battery. Even if outside temperature is let's say 10 degrees Celsius, temperature in the engine compartment is probably higher and could very well be 30 degrees Celsius.