The definition of a mammal is simple: descent from the most recent common ancestor of all mammals. In practice, to avoid circularity, it is sufficient to take the MRCA of a few indisputable mammalian groups such as primates, rodents, carnivorans, ungulates, etc. to include all mammals.

This definition is useful because it turns out that there are many traits unique to mammals, and any given mammal will have almost all these traits. Many such traits are anatomical/biochemical/etc. (Many outwards traits like live birth or so-called "warm blood" aren't unique to mammals.)

However, even if this definition wasn't useful to us, the group Mammalia would still exist. It's a natural evolutionary group (clade) in phylogenetics, to which we merely give a name. (Edit: and cladistics is a natural way of classifying species (among other ways). By natural I mean a classification that tends to match common and unique traits of species in the same clade, and which is causally linked to the history of of the species and to predictions for their future, so that I would expect aliens to have a relatively high probability of using similar classifications.)

The precise clade referred to by the word Mammalia can change depending on context. It makes sense to ask whether borderline species like platypuses are mammals or a sister group of mammals. That's the fuzzy nature of any classification of real things. But the natural limits of the category "mammals" lie somewhere around the monotremes. A group which doesn't include dolphins is definitely not the group of all mammals.