What's the purpose of such weird wiring inside this smart card?
This stripe is a patch antenna, which is nothing else than a wire, printed on a PCB or similar, that radiates like an antenna.

The unconnected strip on the right is likely a reflector element, which doesn't receive any current from a direct connection but is electrically coupled with the rest of the patch to improve the radiation properties or the input impedance (these antennas are very sensitive to resonance).

While there is theory about microstrip (patch) antennas, they're not as easy as dipoles or "normal" antennas to analyze for human beings, anda great part in their design is done numerically. They can easily be considered black magic.

I'm trying to understand If AB are really connected or not:


at a first glance I'd say that tey are not connected, but the rounded areas are the pads to connect the ends of the antenna to the transceiver. It would be consistent with the dark path going to A.
Or, but this may be or may not be: some patch antennas are made of two "independent" radiating elements, of which one or both is feeded by the signal; these elements interact together like a Yagi antenna, providing a more directional radiation. So it also may be possible that there are two different loops, but it's only an hyphotesis.