For decades, Americans have been exposed to a number of ad campaigns that promote an agricultural product or commodity in a generic fashion. Besides "Got Milk?" and "Behold/Ahh, the Power of Cheese," we can also point to "Beef: It's What's for Dinner," "Pork: The Other White Meat," and "The Incredible, Edible Egg." (And outside the grocery department, "Cotton: The Fabric of Our Lives.")

Depending on your perspective, you could say the campaigns were funded and organized by the government, by industry organizations, or by the stakeholders in the industries who were forced to cooperate. (You could also say the consumer is ultimately funding it, since the costs must be passed on.)

Basically, under a series of federal laws starting in 1935, participants in various sectors of agricultural production have been required to make contributions based on their production and import quantities to research and promotion boards which spend those funds on various programs of common benefit. The boards are overseen by the Agricultural Marketing Service, an arm of the U.S. Department of Agriculture (USDA). There are currently something like 21 of these so-called checkoff programs. That name comes from their origin as voluntary programs: stakeholders used to check a box to indicate that they wished to contribute. There are also state-level checkoff programs. At least some of the checkoff programs also receive some taxpayer funding.

Among the things the checkoff programs fund are advertising campaigns designed to stimulate demand for the product as a whole. The rationale here is that most consumers don't differentiate strongly, if at all, between different brands of milk, cheese, beef, eggs, etc., many of which are sold or consumed in a non-branded context. (Last time you paid an extra dollar at a bistro for a cheeseburger or cheese fries, did you ask what brand the cheese was?) In addition, many of the stakeholders are so small that they would not be able to advertise effectively (though this is part of why they form marketing cooperatives). So the most effective strategy for the stakeholders as a whole is to attempt to expand the entire market. The rising tide of commodity consumption should thus lift all boats.

The ads have to be generic. They can't promote any particular brand. They also can't directly disparage any other commodity. Making participation mandatory eliminates concern over free-riders.

The checkoff programs also fund research on the commodity and its uses and efforts to raise its visibility in related industries such as grocery and restaurants. This results in, for example, pizza restaurants using more cheese in their basic recipes, promoting "extra cheese" as a topping, and coming up with even cheesier offerings, like stuffing the crust with cheese.

The specific programs you mentioned are funded by different checkoff programs, since dairy farmers (own cows, get milk out of them) and dairy processers (turn cow's milk into consumer products, ranging from milk in cartons and bottles to cheese to butter to ice cream) participate into different checkoff programs.

"Got Milk?" was actually started by the California Milk Processor Board in 1993, then was licensed by the Milk Processor Education Program (MilkPEP) aka National Fluid Milk Processor Promotion Program in 1995. So that's a state-level checkoff program that went national. MilkPEP is funded by a fee of 20 cents on each hundred pounds of milk sold, which was authorized by the Fluid Milk Promotion Act of 1990.

"Behold/Ahh, The Power of Cheese" was a campaign of the American Dairy Association, which is a name used by Dairy Management, Inc. (DMI), which runs the programing for the dairy farm checkoff program aka National Dairy Promotion & Research Board. The campaign started in 1998. DMI is funded by a fee of 15 cents on each hundred pounds of milk produced and 7.5 cents on each hundred pounds of milk imported. This was authorized by the Dairy Production Stabilization Act of 1983.

The dairy sector's particular problem is that milk is chronically overproduced and consumption of fluid milk (as in, "I'll have a glass of milk") has fallen from historic highs. According to the USDA's Economic Research Service, in 1950 annual per capita consumption of milk and cream was over 325 pound. It dropped to about 275 pounds per year in 1970 and has been hovering a little above 200 since 2000. This decline was seen as a potential crisis in the 1980s. Although milk drinking hasn't recovered, arguably it has held the line, and increased consumption of non-fluid products such as cheese (especially) and butter has aided the industry. (I'm not really qualified to write an economic history of the dairy sector, though!)

The checkoff programs have been subject to legal challenges since the 1990s. Some stakeholders argue that the checkoffs compel them to participate in speech with which they don't agree: a government-compelled subsidy of private speech, as they label it. This would be a First Amendment violation But the courts seems generally inclined to use the category of government speech, which means the First Amendment does not apply.

EDITED to fix some duplication and diary/dairy errors.