Many of the Ancient Greeks would have read the myths as metaphorical tales presenting metaphorical truths about the natural world and the psyche of humanity.
Gods are not physical beings, and so any activity they supposedly did in the physical world clearly is not true. Some will disagree with me on that, but I urge them to look at history, science, and metaphysical theory.
As Lark said, spiritual beings, that is, beings not confined by matter can shift their appearance, or rather, perhaps it is our perception which shifts; or a mixture of the two.
It is scientifically/biologically impossible for a human to physically shift into another animal. Magical practices cannot defy the natural laws.
This said, some folk shift their forms on the astral, there are many magical practicies in which you change your astral and etheric body into something other than your own, sometimes this can be a minor form of invocation. But, the subtle body is subjective and no one would know, your physical body will always be human. You cannot change that with any amount of wishful thinking.