Comparison of humans and other primates in milk production amount
The comparison to non-human primates and the behavioral question kind of threw me, but I think this is an X-Y problem. If you're asking:


  Are humans capable of producing more milk than they need to feed their own offspring


The answer is yes

There is some data I don't think we'd be able to get anymore, but in Chapter 5 of Nutrition During Lactation, they break it down.


  Several studies indicate that potential milk production in humans is considerably higher than the average intake by single infants. Kaucher and colleagues (1945) measured maximum milk output with intrusive and tedious mechanical methods to extract all the mother's milk and reported that production averaged almost 1,200 g/day at 6 to 10 days post partum. This level is much higher than the 500 to 700 g/day consumed by breastfed infants at the same age (Casey et al., 1986; Saint et al., 1984). 


Kaucher's methods may not have been particularly good, because others showed limits more than twice that


  Women who express surplus milk for a milk bank have been shown to produce as much as 3,000 g/day (Macy et al., 1930).


As my wife and many lactating mothers who use a breast pump will be able to confirm, this isn't a relic of the 30s and 40s. If you keep pumping, you keep getting milk.


  In two separate studies, milk production increased by 15 to 40% when a breast pump was used to remove additional milk after feedings (Dewey and Lönnerdal, 1986; Neville and Oliva-Rasbach, 1987).