Generally things like this work better the less you explain them. But since you asked...
The basic idea behind shape-shifting is that the shapes we think are the reality, really aren't. Instead they are reflections or shadows cast by the underlying ideas of the forms. So a shape-shifter is simply something that can cast "shadows" that match multiple other forms. It is not a human and it is not wolf and it is not both, but depending on the situation it appears the same to some arbitrary degree.
This is a very old idea in most parts of the world. It is common to think that there are multiple layers of reality and that the reality we experience is not the real real.
Unfortunately, while the part about what we experience not being the real reality is somewhat supported by science, the difference is not in a form that would support functional shape-shifting without hand-waving. This is not usually a real problem as the precise mechanics of shape-shifting are rarely important.
There is one corner case though that is kind of "scientific" enough for you. It probably won't be compatible with your story otherwise, but maybe it will give a usable idea.
If the reality the characters experience is in actuality a simulation, think Matrix or sci-fi VR game, then it will work more like the theory of shape-shifting requires.
Every character will have attributes that tell what shape it is and there will be absolutely nothing that stops code from changing those attributes at run time. Most likely it would have defined multiple forms and transitions for transferring between them. So it really would work exactly like in fiction. And it would be "real" in the sense that the characters really do experience it.