It’s inaccurate to say that Canada has completely “closed the border to milk” from the US.
While there are restrictions and a long-standing trade dispute regarding dairy products, Canada doesn’t have an outright ban.
The “Got Milk?” controversy and the US-Canada dairy trade dispute center around Canada’s supply management system:
In summary: The issue isn’t a complete closure of the border but rather a long-standing trade disagreement over Canada’s approach to managing its dairy supply and the level of access the US should have to the Canadian market.