Rav Moshe's psak on "Chalav HaCompanies" is restricted to the U.S. and Canada because of the strictness of the FDA inspections and the laws and procedures involving the farms and the dairy processors. It has nothing to do with "common assumptions" as we see in European countries where such milk is not allowed.
Rav Moshe Feinstein’s Heter of Cholov Stam Revisited
Currently, the government inspects all milk farms 2-6 times per year.
Governmental (state) farm inspection protocol specifically includes a
provision that only cows are in the farms’ milking parlors and/or
cowyard. This provision (formulated in terms of swine) is part of the
standard farm inspection form. (See sections of Dairy Farm Inspection
Government inspectors track the intake and output of all milk at
dairies. Thus, the source farms are identified by the inspectors, and
they must correlate with farms approved by the government.
In what countries is chalav yisrael not required for milk? also discusses the matter.
The official definition of milk from the FDA is restricted to cows. "goats milk" is restricted to goats. Changing the definition to include any other animal would require publication in the official FDA "Request for Comments" and require specific reference to the percentages of additional ingredients. Even discussions before the "request for Comments" is issued must first reach the public and the dairy industry.
see the comment with the link from @Aaron
Thus, according to the FDA, cows milk and goats milk may not be mixed without explicit labeling, much less that of any other animal.
There is a movement by lobbyists to include Aspartame (artificial sweetener) in the definition but people are objecting to that. If they would change to definition to include that of other animals, they will have to first publicize the matter first.
If the FDA is going to change the rules, then they would have to give the source animal on each label. Even if the definition changes, it would not allow "blended" milk from different animals. Thus, "Cows milk" must only be from cows. "goats milk" must only be from goats, etc. As a result, one should not worry about milk from nonkosher animals being sold without a reference to the animal it came from.
Senator Dan Coats letter is dealing with mock milk (such as Soy Milk) rather than milk from other animals. If milk from other animals is mixed in to the milk from a dairy, the dairy can be forced to buy all such adulterated products at the full price of Grade A milk. It is that penalty which Rav Moshe used to allow "Chalav HaCompanies".
Code of Federal Regulations Revised as of April 1, 2014 PART 131 -- MILK AND CREAM
(a) Description. Milk is the lacteal secretion, practically free from colostrum, obtained by the complete milking of one or more healthy cows. Milk that is in final package form for beverage use shall have been pasteurized or ultrapasteurized, and shall contain not less than 8 1/4 percent milk solids not fat and not less than 3 1/4 percent milkfat. Milk may have been adjusted by separating part of the milkfat therefrom, or by adding thereto cream, concentrated milk, dry whole milk, skim milk, concentrated skim milk, or nonfat dry milk. Milk may be homogenized.