Is the lateral brake disc position standardized (and, if so, to what tolerance)? I am thinking about getting a second wheel set for my CX bike to be able to easily swap between road and off-road tires (and possibly run a tubeless setup). The bike is equipped with 12mm through-axles (front and back) and flatmount brakes.

I am wondering (assuming same disc diameters): Would the disc rotors be exactly[1] in the same place laterally, or would I have to adjust the brake caliper position after changing the wheels? 
Would it make a difference if I got the same/different brands or models of wheels/hubs?

[1] exactly: close enough, so that the brake pads don't rub on the rotors ;-)
In my experience the answer is kinda yes. You won't need to re-align the calipers with every wheel change, but you will likely need to adjust the brake pads. Hydraulic brakes make the process easier since they are self-adjusting. On bikes with mechanical disk brakes you will likely need to adjust the pads with every wheel change. 

In theory, every 6 bolt rotor and centerlock hub should put the disk in the same place and plane so that this should all just work. In practice the tolerances for disk brakes are so small that even slight differences will cause the disk to rub on the pads. There are two main reasons for this. 

Brake disks aren't flat and don't stay flat over time.

The surface you mount the brake disk on isn't flat. 

Centerlock was created to help with the second problem, they do seem to be a bit better at staying interchangeable than 6 bolt. The only way to overcome the first is to get good at brake disk truing and work to get the disks on your rims all the "same". 

Having said all this, you can get your wheels "good enough" that swapping wheels can be pretty straightforward. But it's unlikely that will happen out of the box and keeping the rotors true enough to keep your wheels swap-able does take some ongoing maintenance.