Well, to your readers it will be magical and they might call it magic regardless of what extremes you go to to portray it as non-magic because in overly simplistic, modern thought, anything beyond the natural realm of possibility can be described as "magical" or "magic". But keep in mind, if/when readers/critics say that about your book, they're saying it as a descriptor, not necessarily as a judgment. So yeah, some readers might consider it magic, but if they're reading a fantasy story, odds are they were hoping for something slightly magical anyway.
Now from the theoretical perspective, what you're describing actually falls more within the realm of what we HWers call "cobha" (pronounced "coat-vaughn). That's just a fancy word that Tsahraf Chasid Mimetes came up with for anything that is natural/possible in a fantasy story that would be impossible in our reality. So in your world, there are people that are naturally born with the ability to turn into bears, which is impossible in our world. That would be cobha.
Now there are times magic can be cobha. For instance if magicians and so on are a natural part of your world, then their powers would be part of that world's cobha. But usually, around here, the distinction is that when HWers say magic in the negative/undesirable context, they're referring to the ways that people try to gain/use supernatural powers that they were not created to have. This falls under the same category as witchcraft and sorcery and so on which God forbids in the Bible. But your idea is nothing like that, so I'm getting off topic and rambling.
Anyways...yeah... Basically, most HWers wouldn't view your shapeshifting idea as magic, but your uninitiated readers might use the broad sense of the word "magic" or "magical" to describe it. But it sounds like a fun idea, so who cares what people call it?
* grins *