Laptop wifi card doesn't look like I expect

The cables are antennas. Both antennas work the same way so connecting them the wrong way around is fine.

If there is some software that can show you from which direction the signal is strongest, that software would end up getting left and right wrong, but that's all there is.*

There is a small chance that the 2 antennas are specced differently (different impedance). Connecting them wrong would not break anything in that case, but signal reception would be below optimal. Again the chance that the antennas have different specs is small. Connecting them the other way around to test is fine.

Also, sadly, it's very common that manuals with pictures (or even without) don't match reality.

*You may have heard that multi antenna systems can boost the signal in the direction of the router. That isn't a problem here. Switching left and right is not a problem even in this case because if the router is on the left, the system will think the router is on the right (switched), and then attempt to boost the signal to the right which ends up boosting the signal correctly to the left (switched again). Just like -1 * -1 = +1.

I see your dilemma. I'd trust the colors and board print over the tags on the wires since they were added by an assembly worker who can easily make a mistake attaching those glued on labels by being in a real hurry to get them on, although the worker and the PDF seem to be in agreement. The screened, firmly mounted label on the card is usually more reliable but it could be what is wrong.

I agree with Peter's answer.