In England we use the word "make" as often, if not more in some areas, as the word "brand". As in "who is it made by" or "what make is it", both asking what brand the product is.
Different meaning. The Duolingo sentence says that they like the brand of toothpaste. You are saying you like the toothpaste of that brand, so not the brand in particular, but just that toothpaste.
I don't agree with your answer. The Dutch sentence doesn't say we like the brand in general, but only when it comes to toothpaste. Maybe we like a totally different brand of toothbrushes. Also note that the English translation doesn't mean "We like the brand of this toothpaste."
I'm not sure whether "from this brand" is proper English, but if it is, AnahitaGupta should report it.
While more specific than DL usually gets, I think l.Harm's point here is that the object of the Dutch sentence is the brand, whereas the object of yours is the toothpaste.
Depending on the context, this could be quite a difference.
Maybe Duo likes Scopsident brand because they support sanctuaries for retired owls, but has no special opinion on the toothpaste itself.
Maybe you like toothpaste from a given brand because it has the Dutch flag in the stripes, but you don't otherwise care about the brand itself - your favourite toothpaste just happens to come from them.
EDIT to mention: "...from this brand" is perfectly correct English, yes. As for whether it's a good translation or not, that really depends on how liberal we wish to be with our translations, and that tends to vary from one course team to another.