Way back when, the stub was always taken on boarding. Even after coupon lift at the gate was introduced, the practice of taking the stub continued.

The remarks area on the stub was used to identify all manner of cabin service issues (SPML, WHCH, CONs, etc.) There used to be special holders in the F galley where the stubs would be put to assist with passenger identification.

Now, all of that information is spit out on the manifest in myriad ways, making the stubs irrelevant. But many FA's, from decades of habit, still tear them.

(It was always weird to me to be left with the stub... I first experienced it when CX went to new style boarding passes in '94.)