You say that, but many systems in the car that rely on timing (ignition, ABS, etc) are run on ECUs that use the 32-bit UNIX epoch for clock-ticks. So they might actually know what day it is, sort of.
As an aside: since this is an unsigned 32-bit integer, many of these systems may become confused at 03:14:08 on 19 January 2038. But only if the programmer(s) used a static start date, instead of "time since the battery was last connected". Since most jobbing programmers are as lazy as I am, there's a non-zero probability that some idiot somewhere has done this.