Despite shortcomings of a survey-based study fraught with patients' dietary recall and diagnosis interpretation biases, the main take-home message is that: while total protein intake correlates with risk for gout, animal proteins are worse than plant proteins.
Attempts to dissect whether some animal proteins are more hazardous than others run into statistical absurdity (like fish and milk being significant only for women but not men), or being contradictory to conventional be...liefs or other observational studies (like fish and poultry are worse than red meat, or that diary is bad).
I also wouldn't sweat it because, even if the associations were true (and it's a super-duper "if"), the clinical significance is miniscule. The hazard ratio of the worse offender (poultry) is 1.27, while that of the surprising saint (legumes) is 0.83; risks hardly worth anyone's effort to modify their diet.
Fructose intake predisposes to the development of hyperuricaemia and gout in both men and women in a dose-dependent manner. Artificially-sweetened soft drinks were not associated with gout.