Milk, in the real world, is an ordinary drink but, in Conker's Bad Fur Day, it's the obsession of the Panther King. All he does is sit on his throne all day and drink it. He only had one problem- the table he set his milk on was missing a leg. Every time he set his milk on the table, it would tip over and the glass holding would break and this would infuriate the king. His obsession with milk even lead him to hire a hitman to kill the girlfriend of the squirrel he wanted as the fourth leg.
In the first chapter, the first cinema shows the Panther King drinking milk but his glass breaks becuase of the problem with his table.
The opening cinematic of Windy shows the Panther King ordering Von Kriplespac to fix the table problem. Later, after Conker heads up to Poo Mountain, Von Kriplespac can be seen showing the Panther King the solution to the table problem. The Panther King then puts a bounty on Conker to be used as a table leg.
The last time milk is mentioned is in the last chapter when Conker and Berri confront the Panther King in his vault. After feeling indegestion, he calls for Von Kriplespac for his milk. Von Kriplespac actually reveals that its because of the King's obssesion with milk that he was able to put an alien egg in the mik (which later birthed Heinrich).