In the United States, no regulatory agency (like the Food and Drug Administration) determines what the phrase “dairy-free” means. It is essential to look at ingredient lists, watching for items like caseinate and whey.
The FDA does regulate the phrase “non-dairy,” which is used for artificial coffee creamers and many other products. Surprisingly, small amounts of caseinate are often allowed under this regulation, making the product off limits for anyone with a milk allergy. Read the label carefully.
Watch out for substitute cheese, notably soy cheese. It often contains casein.
As for milk substitutes, like soy milk, rice milk, and almond milk, most are genuinely dairy free, but of course it pays to read the label and not rely on the phrases non-dairy or dairy-free.
If a product is labeled “vegan” it is supposed to be completely free of all animal products, which includes dairy. It doesn’t hurt to check the ingredient list, however.