I have dim memories of QF using "Captain Cook" as a brand name for some service (maybe the F trans-pacific?). Perhaps the lounges retained that name from the F service branding.
The MEL lounge arrangement is rather mysterious. Especially as the QF lounge is much nicer than the CX lounge. I suspect that the QF lounge was built later and CX either got tired of waiting or it was cheaper to have their own lounge rather than paying QF. SYD is probably too expensive to get lounge space (presuming any would be available, it is rather tight in that terminal).

CX and QF have competed vigorously in the past but seem to be more friendly in the past year. And HKG has recognized that DBX and SIN are big threats to its future, thus the push to bring more flights into HKG before it is too late. Even if it means sleeping with the enemy.