But even beyond that you have the math wrong. Andromeda galaxy is only 2.5 million light-years from Earth. To get a 1 day difference in time dilation across 2.5 million years would take 1 in 913 million difference which someone traveling at even 100MPH (45m/s) isn’t going to see. (1 - (45^2/((3x10^8)^2))^0.5 simply isn’t enough.
I did the calculations from
https://en.wikipedia.org/wiki/Special_relativity#Lorentz_tra... and it comes to ~0.25 years if the person in the car is moving towards Andromeda at 30m/s.
γ is indeed very close to 1, so the humans' watches would still agree closely after 2.5 million years, but the humans would be a quarter of a light year apart at that point, so they need to disagree about something to get the same observed speed of light from Andromeda.
Direction of travel was undefined in the example, just a car on the earth. Suppose they where traveling 30m/s away from andromeda vs towards it suddenly the timing switches.
Chose a specific vector and both observers exactly agree with the events timing. Essentially the only thing they disagree about is what Andromeda’s location and velocity is. (Edited this several times, but it all simplified to choice of vectors.)
But even that’s assuming spaceships rather than “one stationary and one travelling past in a car” the guy in a car is never going to end up light years distant from the stationary person.
But even beyond that you have the math wrong. Andromeda galaxy is only 2.5 million light-years from Earth. To get a 1 day difference in time dilation across 2.5 million years would take 1 in 913 million difference which someone traveling at even 100MPH (45m/s) isn’t going to see. (1 - (45^2/((3x10^8)^2))^0.5 simply isn’t enough.