There's no adjustment of the K factor at the factory. They assume that the calculated fuel flow is the same as the actual fuel flow. The factory default K factor is 1000. It's probably random which MPG is higher when the car is new. Other than that, you pretty much got it!
It's been my experience with two direct-injected BMW's is that the indicated MPG increases over time when compared to the actual MPG. My 2014 535i started out just about perfectly accurate, and over 60k miles the indicated MPG drifted to being about 2.5% more than the actual MPG, meaning the fuel flow became less and less than the computer thought it was. Frau Putzer's 2018 X3 indicated MPG started out about 4% below the actual MPG and drifted to about 3.5% below actual MPG over about 13k miles. Initially, the fuel flow was more than the computer thought it was, but the trend was that the fuel flow was going toward being less than the computer thought.
My port injected 2007 Chevy Cobalt's actual MPG started out about 2% over the indicated MPG. But, after 12 years and 123k miles, it's just about spot on now.
The precise way to measure indicated MPG over several tanks of fuel is to calculate the "on-board computer (OBC) fuel used for each tank, being the miles driven on the tank of fuel divided by the OBC MPG. You then sum up the OBC fuel used for each tank to get the multiple tanks OBC fuel used. With the multiple tanks OBC fuel used, you can calculate the multiple tanks OBC MPG. Then you compare the multiple tanks actual MPG and OBC MPG to determine what to change the K factor to. Using the data for multiple tanks reduces the randomness in actual MPG for a single tank of fuel. That's how I (eventually) determined my new K factors, and got my indicated MPG so close to the actual MPG.
Depending on the particular BMW, the new K factor can be:
K(new) = (OBC MPG / actual MPG) x K(current) (as for Frau Putzer's G01 X3)
K(new) = (actual MPG / OBC MPG) x K(current) (as for my F10 535i)