Scion:
I have been wondering if it is possible to chain 4-digit display so you control them over only 3 wires.
It should be. But first, these displays require you to multiplex them in code - how did you find the code since it is not on the website?
Scion:
Now I have been using 3 digital io-pins / display and with 4 displays it uses 12 digital io-pins.
No, it would need six pins for four displays connected in parallel.
Scion:
There seems to be extra holes at the other of the led's PCB and I have been trying to find out if they are for chaining the displays but I have not found anything concrete about the matter.
They certainly should be.
OK, two approaches to this.
Firstly, these displays have no resistors. This in turn has two consequences; one is that the brightness of individual digits or segments changes according to how many digits or segments are lit at the time. The other is that the 74HC595s are being operated beyond their ratings and may degrade over time and fail either progressively or suddenly.
You can connect them without chaining using six pins, two for all the clock and data pins and one for each latch pin or commoned clock and latch pins and four separate data pins. The second way requires you to use different code, the first requires you only to change the four "instances" of the driver code for the appropriate pins.
Or you should be able to chain them, but again, the code needs to be adjusted to suit and it would indeed require just three pins. Where is your code? (Note the instructions for posting code.)
I would consider the better approach to use properly designed modules using MAX7219s such as these:
These are most certainly designed to be chained to just three control pins and do not require multiplexing in your code. And they are (but controllable in software so that they can be if you choose) much brighter.