Rebranding (or contract manufacturing) has been around much longer than people are giving it credit for. It's a way for a company to sell an expanded line of products even though they don't have the capacity to make all of those types of products. I have a saxophone from the 1960s that was marketed in the US by Olds, a company that mostly made brass (trumpets etc), so rather than build a whole new production line, they bought in a sax from a French maker with "Olds" engraved on it. In photography, we are not as used to this because Kodak made everything from film to cameras to lenses to developing equipment in-house (which ultimately was not so great for Kodak), and Fuji did/does the same thing to a lesser extent.
But you could buy rebranded film in the 1990s, it's just that nobody buying store brand or relabeled color film got bent out of shape about what the film really was. There are other examples in photography like Konica lenses made by Tokina, various brand cameras manufactured by Cosina, etc. The principle is that the rebranding company puts their name and therefore reputation behind the product. The regrettable difference now is that the company putting "Polaroid" on cheap batteries doesn't care about devaluing the Polaroid brand, it is just a name to them where they are mining what is left of market goodwill until the mine is exhausted.
In this case, regardless of which of the two Ilfords is selling the camera, they want to sell a color disposable (I think most people who buy a single use camera want color), but they don't make color film, so put "Ilfocolor" on the front. Even if the film is Kodak, they can't put "Kodak" on the front. Only people on Photrio who are reading tea leaves for film availability will be disappointed that one or the other Ilford didn't magically conjure up their own color film production line.